A Brief Introduction to Tensors and their properties

 

 

 

1. BASIC PROPERTIES OF TENSORS

 

 

1.1 Examples of Tensors

 

The gradient of a vector field is a good example of a second-order tensor.  Visualize a vector field: at every point in space, the field has a vector value u( x 1 , x 2 , x 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa [email protected]@ .  Let G=u [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHhbGaeyypa0Jaey4bIeTaaCyDai [email protected]@  represent the gradient of u.  By definition, G enables you to calculate the change in u when you move from a point x in space to a nearby point at x+dx [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaey4kaSIaamizaiaahIhaaa [email protected]@ :

du=Gdx [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCyDaiabg2da9iaahEeacq [email protected]@

G is a second order tensor.  From this example, we see that when you multiply a vector by a tensor, the result is another vector. 

 

This is a general property of all second order tensors.  A tensor is a linear mapping of a vector onto another vector.  Two examples, together with the vectors they operate on, are:

 

 The stress tensor

t=nσ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH0bGaeyypa0JaaGPaVlaah6gacq [email protected]@

where n is a unit vector normal to a surface, σ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  is the stress tensor and t is the traction vector acting on the surface.

 

 The deformation gradient tensor

dw=Fdx [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaC4Daiabg2da9iaahAeacq [email protected]@

where dx is an infinitesimal line element in an undeformed solid, and dw is the vector representing the deformed line element.

 

 

 

1.2 Matrix representation of a tensor

 

To evaluate and manipulate tensors, we express them as components in a basis, just as for vectors.  We can use the displacement gradient to illustrate how this is done.  Let u( x 1 , x 2 , x 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa [email protected]@  be a vector field, and let G=u [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHhbGaeyypa0Jaey4bIeTaaCyDaa [email protected]@  represent the gradient of u.  Recall the definition of G

du=Gdx [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCyDaiabg2da9iaahEeacq [email protected]@

Now, let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  be a Cartesian basis, and express both du and dx as components.  Then, calculate the components of du in terms of dx using the usual rules of calculus

d u 1 = u 1 x 1 d x 1 + u 1 x 2 d x 2 + u 1 x 3 d x 3 d u 2 = u 2 x 1 d x 1 + u 2 x 2 d x 2 + u 2 x 3 d x 3 d u 3 = u 3 x 1 d x 1 + u 3 x 2 d x 2 + u 3 x 3 d x 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadsgacaWG1bWaaSbaaSqaai aaigdaaeqaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG1bWaaSbaaSqa aiaaigdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabe aaaaGccaWGKbGaamiEamaaBaaaleaacaaIXaaabeaakiabgUcaRmaa laaabaGaeyOaIyRaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiabgk Gi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaamizaiaadIhadaWg aaWcbaGaaGOmaaqabaGccqGHRaWkdaWcaaqaaiabgkGi2kaadwhada WgaaWcbaGaaGymaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaa iodaaeqaaaaakiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaGcba GaamizaiaadwhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqa aiabgkGi2kaadwhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHciITca WG4bWaaSbaaSqaaiaaigdaaeqaaaaakiaadsgacaWG4bWaaSbaaSqa aiaaigdaaeqaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaS qaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaa beaaaaGccaWGKbGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRm aalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIYaaabeaaaOqaaiab gkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaOGaamizaiaadIhada WgaaWcbaGaaG4maaqabaaakeaacaWGKbGaamyDamaaBaaaleaacaaI Zaaabeaakiabg2da9maalaaabaGaeyOaIyRaamyDamaaBaaaleaaca aIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaa aOGaamizaiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkdaWcaa qaaiabgkGi2kaadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiaadsgacaWG4bWaaSbaaS qaaiaaikdaaeqaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSba aSqaaiaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIZa [email protected]@

We could represent this as a matrix product

[ d u 1 d u 2 d u 3 ]=[ u 1 x 1 u 1 x 2 u 1 x 3 u 2 x 1 u 2 x 2 u 2 x 3 u 3 x 1 u 3 x 2 u 3 x 3 ][ d x 1 d x 2 d x 3 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacaWGKb GaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG1bWaaSba aSqaaiaaikdaaeqaaaGcbaGaamizaiaadwhadaWgaaWcbaGaaG4maa qabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaeaafaqabeWadaaa baWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaigdaaeqaaaGcba GaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaaakeaadaWcaaqa aiabgkGi2kaadwhadaWgaaWcbaGaaGymaaqabaaakeaacqGHciITca WG4bWaaSbaaSqaaiaaikdaaeqaaaaaaOqaamaalaaabaGaeyOaIyRa amyDamaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaa WcbaGaaG4maaqabaaaaaGcbaWaaSaaaeaacqGHciITcaWG1bWaaSba aSqaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXa aabeaaaaaakeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaaGOm aaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaaaO qaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIYaaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaaGcbaWaaSaaae aacqGHciITcaWG1bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaaIXaaabeaaaaaakeaadaWcaaqaaiabgkGi2k aadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciITcaWG4bWaaSba aSqaaiaaikdaaeqaaaaaaOqaamaalaaabaGaeyOaIyRaamyDamaaBa aaleaacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaG4m aaqabaaaaaaaaOGaay5waiaaw2faamaadmaabaqbaeqabmqaaaqaai aadsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizaiaadIha daWgaaWcbaGaaGOmaaqabaaakeaacaWGKbGaamiEamaaBaaaleaaca [email protected]@

Alternatively, using index notation

d u i = u i x j d x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaamyDam aaBaaaleaacaWGPbaabeaakiabg2da9maalaaabaGaeyOaIyRaamyD amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcba GaamOAaaqabaaaaOGaamizaiaadIhadaWgaaWcbaGaamOAaaqabaaa [email protected]@

 

From this example we see that G can be represented as a 3×3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  matrix.  The elements of the matrix are known as the components of G in the basis { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@ .  All second order tensors can be represented in this form.  For example, a general second order tensor S could be written as

S[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaam4uamaaBaaaleaacaaIXaGaaGymaaqabaaakeaacaWG tbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaadofadaWgaaWcba GaaGymaiaaiodaaeqaaaGcbaGaam4uamaaBaaaleaacaaIYaGaaGym aaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIYaaabeaaaOqaai aadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaam4uamaaBaaa leaacaaIZaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodaca aIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaG4maiaaiodaaeqaaaaa [email protected]@

You have probably already seen the matrix representation of stress and strain components in introductory courses.

 

Since S can be represented as a matrix, all operations that can be performed on a 3×3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  matrix can also be performed on S.  Examples include sums and products, the transpose, inverse, and determinant.  One can also compute eigenvalues and eigenvectors for tensors, and thus define the log of a tensor, the square root of a tensor, etc.  These tensor operations are summarized below.

 

Note that the numbers S 11 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIXa [email protected]@ , S 12 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYa [email protected]@ , … S 33 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaIZa [email protected]@  depend on the basis { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@ , just as the components of a vector depend on the basis used to represent the vector.  However, just as the magnitude and direction of a vector are independent of the basis, so the properties of a tensor are independent of the basis.  That is to say, if S is a tensor and u is a vector, then the vector

v=Su [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaC4uaiabgwSixl [email protected]@

has the same magnitude and direction, irrespective of the basis used to represent u, v, and S.

 

 

1.3 The difference between a matrix and a tensor

 

If a tensor is a matrix, why is a matrix not the same thing as a tensor?  Well, although you can multiply the three components of a vector u by any 3×3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  matrix,

[ b 1 b 2 b 3 ]=[ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ][ u 1 u 2 u 3 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacaWGIb WaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOyamaaBaaaleaacaaIYaaa beaaaOqaaiaadkgadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaay zxaaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaamyyamaaBaaaleaa caaIXaGaaGymaaqabaaakeaacaWGHbWaaSbaaSqaaiaaigdacaaIYa aabeaaaOqaaiaadggadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGa amyyamaaBaaaleaacaaIYaGaaGymaaqabaaakeaacaWGHbWaaSbaaS qaaiaaikdacaaIYaaabeaaaOqaaiaadggadaWgaaWcbaGaaGOmaiaa iodaaeqaaaGcbaGaamyyamaaBaaaleaacaaIZaGaaGymaaqabaaake aacaWGHbWaaSbaaSqaaiaaiodacaaIYaaabeaaaOqaaiaadggadaWg aaWcbaGaaG4maiaaiodaaeqaaaaaaOGaay5waiaaw2faamaadmaaba qbaeqabmqaaaqaaiaadwhadaWgaaWcbaGaaGymaaqabaaakeaacaWG 1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamyDamaaBaaaleaacaaIZa [email protected]@

the resulting three numbers ( b 1 , b 2 , b 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa [email protected]@  may or may not represent the components of a vector.  If they are the components of a vector, then the matrix represents the components of a tensor A, if not, then the matrix is just an ordinary old matrix.

 

 To check whether ( b 1 , b 2 , b 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa [email protected]@  are the components of a vector, you need to check how ( b 1 , b 2 , b 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa [email protected]@  change due to a change of basis.  That is to say, choose a new basis, calculate the new components of u in this basis, and calculate the new matrix in this basis (the new elements of the matrix will depend on how the matrix was defined.  The elements may or may not change [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  if they don’t, then the matrix cannot be the components of a tensor).  Then, evaluate the matrix product to find a new left hand side, say ( β 1 , β 2 , β 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiabek7aInaaBaaaleaaca aIXaaabeaakiaacYcacqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaGG [email protected]@ .  If  ( β 1 , β 2 , β 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiabek7aInaaBaaaleaaca aIXaaabeaakiaacYcacqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaGG [email protected]@  are related to ( b 1 , b 2 , b 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa [email protected]@  by the same transformation that was used to calculate the new components of u, then ( b 1 , b 2 , b 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa [email protected]@  are the components of a vector, and, therefore, the matrix represents the components of a tensor.

 

1.4 Formal definition

 

Tensors are rather more general objects than the preceding discussion suggests.   There are various ways to define a tensor formally.  One way is the following:

 

* A tensor is a linear vector valued function defined on the set of all vectors

 

More specifically, let S(v) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbGaaiikai [email protected]@  denote a tensor operating on a vector.  Linearity then requires that, for all vectors v,w [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWH2bGaaiilai [email protected]@  and scalars α [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 [email protected]@

·         S(v+w)=S(v)+S(w) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbGaaiikai aahAhacqGHRaWkcaWH3bGaaiykaiabg2da9iaadofacaGGOaGaaCOD [email protected]@

·         S(αv)=αS(v) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbGaaiikai abeg7aHjaahAhacaGGPaGaeyypa0JaeqySdeMaam4uaiaacIcacaWH [email protected]@

 

Alternatively, one can define tensors as sets of numbers that transform in a particular way under a change of coordinate system.  In this case we suppose that n dimensional space can be parameterized by a set of n  real numbers x i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bWaaSbaaS [email protected]@ .   We could change coordinate system by introducing a second set of real numbers x i ( x k ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWG4bGbauaada WgaaWcbaGaamyAaaqabaGccaGGOaGaamiEamaaBaaaleaacaWGRbaa [email protected]@  which are invertible functions of x i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bWaaSbaaS [email protected]@ .   Tensors can then be defined as sets of real numbers that transform in a particular way under this change in coordinate system.  For example

 

·         A tensor of zeroth rank is a scalar that is independent of the coordinate system.

·         A covariant tensor of rank 1 is a vector that transforms as v i = x j x i v j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWG2bGbauaada WgaaWcbaGaamyAaaqabaGccqGH9aqpdaWcaaqaaiabgkGi2kaadIha daWgaaWcbaGaamOAaaqabaaakeaacqGHciITceWG4bGbauaadaWgaa [email protected]@  

·         A contravariant tensor of rank 1 is a vector that transforms as v i = x i x j v j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWG2bGbauaada ahaaWcbeqaaiaadMgaaaGccqGH9aqpdaWcaaqaaiabgkGi2kqadIha gaqbamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaa [email protected]@

·         A covariant tensor of rank 2 transforms as S ij = x i x k x j x l S kl [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGtbGbauaada WgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0ZaaSaaaeaacqGHciIT caWG4bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRabmiEayaafa WaaSbaaSqaaiaadUgaaeqaaaaakmaalaaabaGaeyOaIyRaamiEamaa BaaaleaacaWGQbaabeaaaOqaaiabgkGi2kqadIhagaqbamaaBaaale aacaWGSbaabeaaaaGccaWGtbWaaSbaaSqaaiaadUgacaWGSbaabeaa [email protected]@

·         A contravariant tensor of rank 2 transforms as S kl = x i x k x i x l S ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGtbGbauaada ahaaWcbeqaaiaadUgacaWGSbaaaOGaeyypa0ZaaSaaaeaacqGHciIT ceWG4bGbauaadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG4b WaaSbaaSqaaiaadUgaaeqaaaaakmaalaaabaGaeyOaIyRabmiEayaa faWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaale aacaWGSbaabeaaaaGccaWGtbWaaWbaaSqabeaacaWGPbGaamOAaaaa [email protected]@

·         A mixed tensor of rank 2 transforms as S j i = x k x i x j x l S l k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGtbGbauaada qhaaWcbaGaamOAaaqaaiaadMgaaaGccqGH9aqpdaWcaaqaaiabgkGi 2kqadIhagaqbamaaBaaaleaacaWGRbaabeaaaOqaaiabgkGi2kaadI hadaWgaaWcbaGaamyAaaqabaaaaOWaaSaaaeaacqGHciITcaWG4bWa aSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIyRabmiEayaafaWaaSbaaS qaaiaadYgaaeqaaaaakiaadofadaqhaaWcbaGaamiBaaqaaiaadUga [email protected]@

 

Higher rank tensors can be defined in similar ways.  In solid and fluid mechanics we nearly always use Cartesian tensors, (i.e. we work with the components of tensors in a Cartesian coordinate system) and this level of generality is not needed (and is rather mysterious).  We might occasionally use a curvilinear coordinate system, in which we do express tensors in terms of covariant or contravariant components [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  this gives some sense of what these quantities mean.   But since solid and fluid mechanics live in Euclidean space we don’t see some of the subtleties that arise, e.g. in the theory of general relativity.

 

 

1.5 Creating a tensor using a dyadic product of two vectors.

 

Let a and b be two vectors.  The dyadic product of a and b is a second order tensor S denoted by

S=ab S ij = a i b j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyypa0JaaCyyaiabgEPiel aahkgacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaam4uamaaBaaaleaacaWGPbGaamOAaaqaba GccqGH9aqpcaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaamOyamaaBaaa [email protected]@ .

with the property

Su=( ab )u=a(bu) S ij u j =( a i b k ) u k = a i ( b k u k ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyXICTaaCyDaiabg2da9m aabmaabaGaaCyyaiabgEPielaahkgaaiaawIcacaGLPaaacqGHflY1 caWH1bGaeyypa0JaaCyyaiaacIcacaWHIbGaeyyXICTaaCyDaiaacM cacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaaSbaaSqaaiaadMgaca WGQbaabeaakiaadwhadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaGG OaGaamyyamaaBaaaleaacaWGPbaabeaakiaadkgadaWgaaWcbaGaam 4AaaqabaGccaGGPaGaamyDamaaBaaaleaacaWGRbaabeaakiabg2da 9iaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamOyamaaBaaale aacaWGRbaabeaakiaadwhadaWgaaWcbaGaam4AaaqabaGccaGGPaaa [email protected]@

for all vectors u.  (Clearly, this maps u onto a vector parallel to a with magnitude | a |( bu ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiaahggaaiaawEa7caGLiW [email protected]@  )

 

The components of ab [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  in a basis { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  are

[ a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 a 3 b 1 a 3 b 2 a 3 b 3 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGHb WaaSbaaSqaaiaaigdaaeqaaOGaamOyamaaBaaaleaacaaIXaaabeaa aOqaaiaadggadaWgaaWcbaGaaGymaaqabaGccaWGIbWaaSbaaSqaai aaikdaaeqaaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaadkga daWgaaWcbaGaaG4maaqabaaakeaacaWGHbWaaSbaaSqaaiaaikdaae qaaOGaamOyamaaBaaaleaacaaIXaaabeaaaOqaaiaadggadaWgaaWc baGaaGOmaaqabaGccaWGIbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaam yyamaaBaaaleaacaaIYaaabeaakiaadkgadaWgaaWcbaGaaG4maaqa baaakeaacaWGHbWaaSbaaSqaaiaaiodaaeqaaOGaamOyamaaBaaale aacaaIXaaabeaaaOqaaiaadggadaWgaaWcbaGaaG4maaqabaGccaWG IbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamyyamaaBaaaleaacaaIZa aabeaakiaadkgadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaayzx [email protected]@

 

Note that not all tensors can be constructed using a dyadic product of only two vectors (this is because ( ab )u [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaahggacqGHxkcXcaWHIb [email protected]@  always has to be parallel to a, and therefore the representation cannot map a vector onto an arbitrary vector).  However, if a, b, and c are three independent vectors (i.e. no two of them are parallel) then all tensors can be constructed as a sum of scalar multiples of the nine possible dyadic products of these vectors. 

 

 

2. OPERATIONS ON SECOND ORDER TENSORS

 

 Tensor components

 

Let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@  be a Cartesian basis, and let S be a second order tensor.  The components of S in { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@  may be represented as a matrix

[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb WaaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadofadaWgaaWcbaGa aGymaiaaikdaaeqaaaGcbaGaam4uamaaBaaaleaacaaIXaGaaG4maa qabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIXaaabeaaaOqaaiaa dofadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaam4uamaaBaaale aacaaIYaGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaI XaaabeaaaOqaaiaadofadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcba Gaam4uamaaBaaaleaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzx [email protected]@

where

S 11 = e 1 ( S e 1 ), S 12 = e 1 ( S e 2 ), S 11 = e 1 ( S e 3 ), S 21 = e 2 ( S e 1 ), S 22 = e 2 ( S e 2 ), S 21 = e 2 ( S e 3 ), S 31 = e 3 ( S e 1 ), S 32 = e 3 ( S e 2 ), S 31 = e 3 ( S e 3 ), [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadofadaWgaaWcbaGaaGymai aaigdaaeqaaOGaeyypa0JaaCyzamaaBaaaleaacaaIXaaabeaakiab gwSixpaabmaabaGaaC4uaiabgwSixlaahwgadaWgaaWcbaGaaGymaa qabaaakiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtb WaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iaahwgadaWgaaWc baGaaGymaaqabaGccqGHflY1daqadaqaaiaahofacqGHflY1caWHLb WaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaaiilaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaadofadaWgaaWcbaGaaGymaiaaigdaaeqa aOGaeyypa0JaaCyzamaaBaaaleaacaaIXaaabeaakiabgwSixpaabm aabaGaaC4uaiabgwSixlaahwgadaWgaaWcbaGaaG4maaqabaaakiaa wIcacaGLPaaacaGGSaaabaGaam4uamaaBaaaleaacaaIYaGaaGymaa qabaGccqGH9aqpcaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaeyyXIC9a aeWaaeaacaWHtbGaeyyXICTaaCyzamaaBaaaleaacaaIXaaabeaaaO GaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadofadaWgaa WcbaGaaGOmaiaaikdaaeqaaOGaeyypa0JaaCyzamaaBaaaleaacaaI YaaabeaakiabgwSixpaabmaabaGaaC4uaiabgwSixlaahwgadaWgaa WcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGccqGH 9aqpcaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaeyyXIC9aaeWaaeaaca WHtbGaeyyXICTaaCyzamaaBaaaleaacaaIZaaabeaaaOGaayjkaiaa wMcaaiaacYcaaeaacaWGtbWaaSbaaSqaaiaaiodacaaIXaaabeaaki abg2da9iaahwgadaWgaaWcbaGaaG4maaqabaGccqGHflY1daqadaqa aiaahofacqGHflY1caWHLbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOa GaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaBaaaleaaca aIZaGaaGOmaaqabaGccqGH9aqpcaWHLbWaaSbaaSqaaiaaiodaaeqa aOGaeyyXIC9aaeWaaeaacaWHtbGaeyyXICTaaCyzamaaBaaaleaaca aIYaaabeaaaOGaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWGtbWaaSbaaSqaaiaaiodacaaIXaaabeaakiabg2da9iaa hwgadaWgaaWcbaGaaG4maaqabaGccqGHflY1daqadaqaaiaahofacq GHflY1caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaGa [email protected]@

 

The representation of a tensor in terms of its components can also be expressed in dyadic form as

S= j=1 3 i=1 3 S ij e i e j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahofacqGH9aqpdaaeWbqaamaaqahaba Gaam4uamaaBaaaleaacaWGPbGaamOAaaqabaGccaWHLbWaaSbaaSqa aiaadMgaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacaWGQbaabeaaae aacaWGPbGaeyypa0JaaGymaaqaaiaaiodaa0GaeyyeIuoaaSqaaiaa [email protected]@

This representation is particularly convenient when using polar coordinates, or when using a general non-orthogonal coordinate system.

 

 Addition

Let S and T be two tensors.  Then U=S+T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHvbGaeyypa0JaaC4uaiabgUcaRi [email protected]@  is also a tensor.

 

Denote the Cartesian components of U, S and T by matrices as defined above.  The components of U are then related to the components of S and T by

[ U 11 U 12 U 13 U 21 U 22 U 23 U 31 U 32 U 33 ]=[ S 11 + T 11 S 12 + T 12 S 13 + T 13 S 21 + T 21 S 22 + T 22 S 23 + T 23 S 31 + T 31 S 32 + T 32 S 33 + T 33 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGvb WaaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadwfadaWgaaWcbaGa aGymaiaaikdaaeqaaaGcbaGaamyvamaaBaaaleaacaaIXaGaaG4maa qabaaakeaacaWGvbWaaSbaaSqaaiaaikdacaaIXaaabeaaaOqaaiaa dwfadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaamyvamaaBaaale aacaaIYaGaaG4maaqabaaakeaacaWGvbWaaSbaaSqaaiaaiodacaaI XaaabeaaaOqaaiaadwfadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcba GaamyvamaaBaaaleaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzx aaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaam4uamaaBaaaleaaca aIXaGaaGymaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaigdacaaI XaaabeaaaOqaaiaadofadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaey 4kaSIaamivamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWa aSbaaSqaaiaaigdacaaIZaaabeaakiabgUcaRiaadsfadaWgaaWcba GaaGymaiaaiodaaeqaaaGcbaGaam4uamaaBaaaleaacaaIYaGaaGym aaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaikdacaaIXaaabeaaaO qaaiaadofadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaey4kaSIaamiv amaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacaWGtbWaaSbaaSqaai aaikdacaaIZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGOmaiaa iodaaeqaaaGcbaGaam4uamaaBaaaleaacaaIZaGaaGymaaqabaGccq GHRaWkcaWGubWaaSbaaSqaaiaaiodacaaIXaaabeaaaOqaaiaadofa daWgaaWcbaGaaG4maiaaikdaaeqaaOGaey4kaSIaamivamaaBaaale aacaaIZaGaaGOmaaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaI ZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaG4maiaaiodaaeqaaa [email protected]@

In index notation we would write

U ij = S ij + T ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaSbaaS qaaiaadMgacaWGQbaabeaakiabg2da9iaadofadaWgaaWcbaGaamyA aiaadQgaaeqaaOGaey4kaSIaamivamaaBaaaleaacaWGPbGaamOAaa [email protected]@

 

 Product of a tensor and a vector

 

Let u be a vector and S a second order tensor.  Then

v=Su [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaC4uaiabgwSixl [email protected]@

is a vector. 

 

Let ( u 1 , u 2 , u 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaadwhadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaamyDamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  and ( v 1 , v 2 , v 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaadAhadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaamODamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  denote the components of vectors u and v in a Cartesian basis { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@ , and denote the Cartesian components of S as described above.  Then

[ v 1 v 2 v 3 ]=[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ][ u 1 u 2 u 3 ]=[ S 11 u 1 + S 12 u 2 + S 13 u 3 S 21 u 1 + S 22 u 2 + S 23 u 3 S 31 u 1 + S 32 u 2 + S 33 u 3 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacaWG2b WaaSbaaSqaaiaaigdaaeqaaaGcbaGaamODamaaBaaaleaacaaIYaaa beaaaOqaaiaadAhadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaay zxaaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaam4uamaaBaaaleaa caaIXaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYa aabeaaaOqaaiaadofadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGa am4uamaaBaaaleaacaaIYaGaaGymaaqabaaakeaacaWGtbWaaSbaaS qaaiaaikdacaaIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaGOmaiaa iodaaeqaaaGcbaGaam4uamaaBaaaleaacaaIZaGaaGymaaqabaaake aacaWGtbWaaSbaaSqaaiaaiodacaaIYaaabeaaaOqaaiaadofadaWg aaWcbaGaaG4maiaaiodaaeqaaaaaaOGaay5waiaaw2faamaadmaaba qbaeqabmqaaaqaaiaadwhadaWgaaWcbaGaaGymaaqabaaakeaacaWG 1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamyDamaaBaaaleaacaaIZa aabeaaaaaakiaawUfacaGLDbaacqGH9aqpdaWadaqaauaabeqadeaa aeaacaWGtbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaadwhadaWgaa WcbaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaigdacaaI YaaabeaakiaadwhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGtb WaaSbaaSqaaiaaigdacaaIZaaabeaakiaadwhadaWgaaWcbaGaaG4m aaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIXaaabeaakiaadw hadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaa ikdacaaIYaaabeaakiaadwhadaWgaaWcbaGaaGOmaaqabaGccqGHRa WkcaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaakiaadwhadaWgaaWc baGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaIXaaabe aakiaadwhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSba aSqaaiaaiodacaaIYaaabeaakiaadwhadaWgaaWcbaGaaGOmaaqaba GccqGHRaWkcaWGtbWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadwha [email protected]@

Alternatively, using index notation

v i = S ij u j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaSbaaS qaaiaadMgaaeqaaOGaeyypa0Jaam4uamaaBaaaleaacaWGPbGaamOA [email protected]@

 

 

The product

v=uS [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaCyDaiabgwSixl [email protected]@

is also a vector.  In component form

[ v 1 v 2 v 3 ]=[ u 1 u 2 u 3 ][ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ]=[ u 1 S 11 + u 2 S 21 + u 3 S 31 u 1 S 12 + u 2 S 22 + u 3 S 32 u 1 S 13 + u 2 S 23 + u 3 S 33 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqabmaaaeaacaWG2b WaaSbaaSqaaiaaigdaaeqaaaGcbaGaamODamaaBaaaleaacaaIYaaa beaaaOqaaiaadAhadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaay zxaaGaeyypa0ZaamWaaeaafaqabeqadaaabaGaamyDamaaBaaaleaa caaIXaaabeaaaOqaaiaadwhadaWgaaWcbaGaaGOmaaqabaaakeaaca WG1bWaaSbaaSqaaiaaiodaaeqaaaaaaOGaay5waiaaw2faamaadmaa baqbaeqabmWaaaqaaiaadofadaWgaaWcbaGaaGymaiaaigdaaeqaaa GcbaGaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWa aSbaaSqaaiaaigdacaaIZaaabeaaaOqaaiaadofadaWgaaWcbaGaaG OmaiaaigdaaeqaaaGcbaGaam4uamaaBaaaleaacaaIYaGaaGOmaaqa baaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiaado fadaWgaaWcbaGaaG4maiaaigdaaeqaaaGcbaGaam4uamaaBaaaleaa caaIZaGaaGOmaaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaIZa aabeaaaaaakiaawUfacaGLDbaacqGH9aqpdaWadaqaauaabeqadeaa aeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaam4uamaaBaaaleaaca aIXaGaaGymaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaaikdaaeqa aOGaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkcaWG1b WaaSbaaSqaaiaaiodaaeqaaOGaam4uamaaBaaaleaacaaIZaGaaGym aaqabaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaam4uamaaBa aaleaacaaIXaGaaGOmaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaa ikdaaeqaaOGaam4uamaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHRa WkcaWG1bWaaSbaaSqaaiaaiodaaeqaaOGaam4uamaaBaaaleaacaaI ZaGaaGOmaaqabaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaam 4uamaaBaaaleaacaaIXaGaaG4maaqabaGccqGHRaWkcaWG1bWaaSba aSqaaiaaikdaaeqaaOGaam4uamaaBaaaleaacaaIYaGaaG4maaqaba GccqGHRaWkcaWG1bWaaSbaaSqaaiaaiodaaeqaaOGaam4uamaaBaaa [email protected]@

or

v i = u j S ji [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaSbaaS qaaiaadMgaaeqaaOGaeyypa0JaamyDamaaBaaaleaacaWGQbaabeaa [email protected]@

Observe that uSSu [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyyXICTaaC4uaiabgcMi5k [email protected]@  (unless S is symmetric).

 

 

 Product of two tensors

 

Let T and S be two second order tensors.  Then U=TS [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHvbGaeyypa0JaaCivaiabgwSixl [email protected]@  is also a tensor.

 

Denote the components of U, S and T by 3×3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  matrices.  Then,

[ U 11 U 12 U 13 U 21 U 22 U 23 U 31 U 32 U 33 ]=[ T 11 T 12 T 13 T 21 T 22 T 23 T 31 T 32 T 33 ][ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] =[ T 11 S 11 + T 12 S 21 + T 13 S 31 T 11 S 12 + T 12 S 22 + T 13 S 32 T 11 S 13 + T 12 S 23 + T 13 S 33 T 21 S 11 + T 22 S 21 + T 23 S 31 T 21 S 12 + T 22 S 22 + T 23 S 32 T 21 S 12 + T 22 S 22 + T 23 S 32 T 31 S 11 + T 32 S 21 + T 33 S 31 T 31 S 12 + T 32 S 22 + T 33 S 32 T 31 S 13 + T 32 S 23 + T 33 S 33 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaadmaabaqbaeqabmWaaaqaai aadwfadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaamyvamaaBaaa leaacaaIXaGaaGOmaaqabaaakeaacaWGvbWaaSbaaSqaaiaaigdaca aIZaaabeaaaOqaaiaadwfadaWgaaWcbaGaaGOmaiaaigdaaeqaaaGc baGaamyvamaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacaWGvbWaaS baaSqaaiaaikdacaaIZaaabeaaaOqaaiaadwfadaWgaaWcbaGaaG4m aiaaigdaaeqaaaGcbaGaamyvamaaBaaaleaacaaIZaGaaGOmaaqaba aakeaacaWGvbWaaSbaaSqaaiaaiodacaaIZaaabeaaaaaakiaawUfa caGLDbaacqGH9aqpdaWadaqaauaabeqadmaaaeaacaWGubWaaSbaaS qaaiaaigdacaaIXaaabeaaaOqaaiaadsfadaWgaaWcbaGaaGymaiaa ikdaaeqaaaGcbaGaamivamaaBaaaleaacaaIXaGaaG4maaqabaaake aacaWGubWaaSbaaSqaaiaaikdacaaIXaaabeaaaOqaaiaadsfadaWg aaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaamivamaaBaaaleaacaaIYa GaaG4maaqabaaakeaacaWGubWaaSbaaSqaaiaaiodacaaIXaaabeaa aOqaaiaadsfadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcbaGaamivam aaBaaaleaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzxaaWaamWa aeaafaqabeWadaaabaGaam4uamaaBaaaleaacaaIXaGaaGymaaqaba aakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaadofa daWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaam4uamaaBaaaleaaca aIYaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIYaaa beaaaOqaaiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaam 4uamaaBaaaleaacaaIZaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqa aiaaiodacaaIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaG4maiaaio daaeqaaaaaaOGaay5waiaaw2faaaqaaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8Uaeyypa0ZaamWaaeaafaqabeWadaaabaGaamivamaaBa aaleaacaaIXaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaigdacaaI XaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGymaiaaikdaaeqaaO Gaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkcaWGubWa aSbaaSqaaiaaigdacaaIZaaabeaakiaadofadaWgaaWcbaGaaG4mai aaigdaaeqaaaGcbaGaamivamaaBaaaleaacaaIXaGaaGymaaqabaGc caWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgUcaRiaadsfada WgaaWcbaGaaGymaiaaikdaaeqaaOGaam4uamaaBaaaleaacaaIYaGa aGOmaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaigdacaaIZaaabe aakiaadofadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcbaGaamivamaa BaaaleaacaaIXaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaigdaca aIZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGymaiaaikdaaeqa aOGaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGHRaWkcaWGub WaaSbaaSqaaiaaigdacaaIZaaabeaakiaadofadaWgaaWcbaGaaG4m aiaaiodaaeqaaaGcbaGaamivamaaBaaaleaacaaIYaGaaGymaaqaba GccaWGtbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaadsfa daWgaaWcbaGaaGOmaiaaikdaaeqaaOGaam4uamaaBaaaleaacaaIYa GaaGymaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaikdacaaIZaaa beaakiaadofadaWgaaWcbaGaaG4maiaaigdaaeqaaaGcbaGaamivam aaBaaaleaacaaIYaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaigda caaIYaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGOmaiaaikdaae qaaOGaam4uamaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHRaWkcaWG ubWaaSbaaSqaaiaaikdacaaIZaaabeaakiaadofadaWgaaWcbaGaaG 4maiaaikdaaeqaaaGcbaGaamivamaaBaaaleaacaaIYaGaaGymaaqa baGccaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgUcaRiaads fadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaam4uamaaBaaaleaacaaI YaGaaGOmaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaikdacaaIZa aabeaakiaadofadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcbaGaamiv amaaBaaaleaacaaIZaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaig dacaaIXaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaG4maiaaikda aeqaaOGaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkca WGubWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadofadaWgaaWcbaGa aG4maiaaigdaaeqaaaGcbaGaamivamaaBaaaleaacaaIZaGaaGymaa qabaGccaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgUcaRiaa dsfadaWgaaWcbaGaaG4maiaaikdaaeqaaOGaam4uamaaBaaaleaaca aIYaGaaGOmaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaiodacaaI ZaaabeaakiaadofadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcbaGaam ivamaaBaaaleaacaaIZaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaa igdacaaIZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaG4maiaaik daaeqaaOGaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGHRaWk caWGubWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadofadaWgaaWcba [email protected]@

Alternatively, using index notation

U ij = T ik S kj [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaSbaaS qaaiaadMgacaWGQbaabeaakiabg2da9iaadsfadaWgaaWcbaGaamyA [email protected]@

 

Note that tensor products, like matrix products, are not commutative; i.e. TSST [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaeyyXICTaaC4uaiabgcMi5k [email protected]@

 

 

 Transpose

 

Let S be a tensor.  The transpose of S is denoted by S T [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  and is defined so that

u S T =Su [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyyXICTaaC4uamaaCaaale [email protected]@

 

Denote the components of S by a 3x3 matrix.  The components of  S T [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  are then

S T [ S 11 S 21 S 31 S 12 S 22 S 32 S 13 S 23 S 33 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbWaaWbaaSqabeaacaWGubaaaO GaeyyyIO7aamWaaeaafaqabeWadaaabaGaam4uamaaBaaaleaacaaI XaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIXaaabe aaaOqaaiaadofadaWgaaWcbaGaaG4maiaaigdaaeqaaaGcbaGaam4u amaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWaaSbaaSqaai aaikdacaaIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaG4maiaaikda aeqaaaGcbaGaam4uamaaBaaaleaacaaIXaGaaG4maaqabaaakeaaca WGtbWaaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiaadofadaWgaaWc [email protected]@

i.e. the rows and columns of the matrix are switched.


Note that, if A and B are two tensors, then

( AB ) T = B T A T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaahgeacqGHflY1caWHcb aacaGLOaGaayzkaaWaaWbaaSqabeaacaWGubaaaOGaeyypa0JaaCOq amaaCaaaleqabaGaamivaaaakiabgwSixlaahgeadaahaaWcbeqaai [email protected]@

 


 Trace

 

Let S be a tensor, and denote the components of S by a 3×3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  matrix.  The trace of S is denoted by tr(S) or trace(S), and can be computed by summing the diagonals of the matrix of components

trace( S )= S 11 + S 22 + S 33 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqG0bGaaeOCaiaabggacaqGJbGaae yzamaabmaabaGaaC4uaaGaayjkaiaawMcaaiabg2da9iaadofadaWg aaWcbaGaaGymaiaaigdaaeqaaOGaey4kaSIaam4uamaaBaaaleaaca aIYaGaaGOmaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaiodacaaI [email protected]@

More formally, let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  be any Cartesian basis.  Then

trace( S )= e 1 S e 1 + e 2 S e 2 + e 3 S e 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqG0bGaaeOCaiaabggacaqGJbGaae yzamaabmaabaGaaC4uaaGaayjkaiaawMcaaiabg2da9iaahwgadaWg aaWcbaGaaGymaaqabaGccqGHflY1caWHtbGaeyyXICTaaCyzamaaBa aaleaacaaIXaaabeaakiabgUcaRiaahwgadaWgaaWcbaGaaGOmaaqa baGccqGHflY1caWHtbGaeyyXICTaaCyzamaaBaaaleaacaaIYaaabe aakiabgUcaRiaahwgadaWgaaWcbaGaaG4maaqabaGccqGHflY1caWH [email protected]@

The trace of a tensor is an example of an invariant of the tensor [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  you get the same value for trace(S) whatever basis you use to define the matrix of components of S.

 

In index notation, the trace is written S kk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS [email protected]@

 

 Contraction.

 

Inner Product: Let S and T be two second order tensors.  The inner product of S and T is a scalar, denoted by S:T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@ .  Represent S and T by their components in a basis.  Then

S:T= S 11 T 11 + S 12 T 12 + S 13 T 13 + S 21 T 21 + S 22 T 22 + S 23 T 23 + S 31 T 31 + S 32 T 32 + S 33 T 33 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahofacaGG6aGaaCivaiabg2 da9iaadofadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaamivamaaBaaa leaacaaIXaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaig dacaaIYaaabeaakiaadsfadaWgaaWcbaGaaGymaiaaikdaaeqaaOGa ey4kaSIaam4uamaaBaaaleaacaaIXaGaaG4maaqabaGccaWGubWaaS baaSqaaiaaigdacaaIZaaabeaaaOqaaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWkca WGtbWaaSbaaSqaaiaaikdacaaIXaaabeaakiaadsfadaWgaaWcbaGa aGOmaiaaigdaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaaIYaGaaG OmaaqabaGccaWGubWaaSbaaSqaaiaaikdacaaIYaaabeaakiabgUca RiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaamivamaaBaaale aacaaIYaGaaG4maaqabaaakeaacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaey4kaSIaam4uam aaBaaaleaacaaIZaGaaGymaaqabaGccaWGubWaaSbaaSqaaiaaioda caaIXaaabeaakiabgUcaRiaadofadaWgaaWcbaGaaG4maiaaikdaae qaaOGaamivamaaBaaaleaacaaIZaGaaGOmaaqabaGccqGHRaWkcaWG tbWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadsfadaWgaaWcbaGaaG [email protected]@

In index notation S:T S ij T ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbGaaiOoai aahsfacqGHHjIUcaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaa [email protected]@

 

Observe that S:T=T:S [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaaiOoaiaahsfacqGH9aqpca [email protected]@ , and also that S:I=trace(S) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaaiOoaiaahMeacqGH9aqpca qG0bGaaeOCaiaabggacaqGJbGaaeyzaiaabIcacaWHtbGaaeykaaaa @[email protected] , where I is the identity tensor.

 Outer product: Let S and T be two second order tensors.  The outer product of S and T is a scalar, denoted by ST [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyXICTaeyyXICTaaCivaa [email protected]@ .  Represent S and T by their components in a basis.  Then

ST= S 11 T 11 + S 21 T 12 + S 31 T 13 + S 12 T 21 + S 22 T 22 + S 32 T 23 + S 13 T 31 + S 23 T 32 + S 33 T 33 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahofacqGHflY1cqGHflY1ca WHubGaeyypa0Jaam4uamaaBaaaleaacaaIXaGaaGymaaqabaGccaWG ubWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaadofadaWgaa WcbaGaaGOmaiaaigdaaeqaaOGaamivamaaBaaaleaacaaIXaGaaGOm aaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaiodacaaIXaaabeaaki aadsfadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abgUcaRiaadofadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaamivamaa BaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaai aaikdacaaIYaaabeaakiaadsfadaWgaaWcbaGaaGOmaiaaikdaaeqa aOGaey4kaSIaam4uamaaBaaaleaacaaIZaGaaGOmaaqabaGccaWGub WaaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRa WkcaWGtbWaaSbaaSqaaiaaigdacaaIZaaabeaakiaadsfadaWgaaWc baGaaG4maiaaigdaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaaIYa GaaG4maaqabaGccaWGubWaaSbaaSqaaiaaiodacaaIYaaabeaakiab gUcaRiaadofadaWgaaWcbaGaaG4maiaaiodaaeqaaOGaamivamaaBa [email protected]@

In index notation ST S ij T ji [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbGaeyyXIC TaeyyXICTaaCivaiabggMi6kaadofadaWgaaWcbaGaamyAaiaadQga [email protected]@

 

 

Observe that ST= S T :T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyXICTaeyyXICTaaCivai abg2da9iaahofadaahaaWcbeqaaiaadsfaaaGccaGG6aGaaCivaaaa @[email protected]

 

 Determinant

 

The determinant of a tensor is defined as the determinant of the matrix of its components in a basis.  For a second order tensor

detS=det[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] = S 11 ( S 22 S 33 S 23 S 32 )+ S 12 ( S 23 S 31 S 21 S 33 )+ S 13 ( S 21 S 32 S 31 S 22 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiGacsgacaGGLbGaaiiDaiaaho facqGH9aqpciGGKbGaaiyzaiaacshadaWadaabaeqabaGaam4uamaa BaaaleaacaaIXaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caWGtb WaaSbaaSqaaiaaigdacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7caWGtbWaaSbaaSqaaiaaigdacaaIZaaabeaaaOqaaiaadofada WgaaWcbaGaaGOmaiaaigdaaeqaaOGaaGPaVlaaykW7caaMc8Uaam4u amaaBaaaleaacaaIYaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7ca WGtbWaaSbaaSqaaiaaikdacaaIZaaabeaakiaaykW7aeaacaWGtbWa aSbaaSqaaiaaiodacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaado fadaWgaaWcbaGaaG4maiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8Ua am4uamaaBaaaleaacaaIZaGaaG4maaqabaaaaOGaay5waiaaw2faaa qaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeyypa0Jaam 4uamaaBaaaleaacaaIXaGaaGymaaqabaGccaGGOaGaam4uamaaBaaa leaacaaIYaGaaGOmaaqabaGccaWGtbWaaSbaaSqaaiaaiodacaaIZa aabeaakiabgkHiTiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGa am4uamaaBaaaleaacaaIZaGaaGOmaaqabaGccaGGPaGaey4kaSIaam 4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccaGGOaGaam4uamaaBaaa leaacaaIYaGaaG4maaqabaGccaWGtbWaaSbaaSqaaiaaiodacaaIXa aabeaakiabgkHiTiaadofadaWgaaWcbaGaaGOmaiaaigdaaeqaaOGa am4uamaaBaaaleaacaaIZaGaaG4maaqabaGccaGGPaGaey4kaSIaam 4uamaaBaaaleaacaaIXaGaaG4maaqabaGccaGGOaGaam4uamaaBaaa leaacaaIYaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaiodacaaIYa aabeaakiabgkHiTiaadofadaWgaaWcbaGaaG4maiaaigdaaeqaaOGa [email protected]@

 

 

 

In index notation this would read

det(S)= 1 6 ijk lmn S li S mj S nk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC 4uaiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI2aaaaiabgIGi opaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaeyicI48aaSbaaS qaaiaadYgacaWGTbGaamOBaaqabaGccaWGtbWaaSbaaSqaaiaadYga caWGPbaabeaakiaadofadaWgaaWcbaGaamyBaiaadQgaaeqaaOGaam [email protected]@

 

Note that if S and T are two tensors, then

det(S)=det( S T )det(ST)=det(S)det(T) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC 4uaiaacMcacqGH9aqpciGGKbGaaiyzaiaacshadaqadaqaaiaahofa daahaaWcbeqaaiaadsfaaaaakiaawIcacaGLPaaacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaciizaiaacwgacaGG0bGaaiikaiaahofacq GHflY1caWHubGaaiykaiabg2da9iGacsgacaGGLbGaaiiDaiaacIca caWHtbGaaiykaiGacsgacaGGLbGaaiiDaiaacIcacaWHubGaaiykaa [email protected]@

 

 Inverse

 

Let S be a second order tensor.  The inverse of S exists if and only if det(S)0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC [email protected]@ , and is defined by

S 1 S=I [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbWaaWbaaSqabeaacqGHsislca [email protected]@

where S 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahofadaahaa [email protected]@  denotes the inverse of S and I is the identity tensor.

 

The inverse of a tensor may be computed by calculating the inverse of the matrix of its components.  Formally, the inverse of a second order tensor can be written in a simple form using index notation as

S ji 1 = 1 2det(S) ipq jkl S pk S ql [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaa0baaS qaaiaadQgacaWGPbaabaGaeyOeI0IaaGymaaaakiabg2da9maalaaa baGaaGymaaqaaiaaikdaciGGKbGaaiyzaiaacshacaGGOaGaaC4uai aacMcaaaGaeyicI48aaSbaaSqaaiaadMgacaWGWbGaamyCaaqabaGc cqGHiiIZdaWgaaWcbaGaamOAaiaadUgacaWGSbaabeaakiaadofada WgaaWcbaGaamiCaiaadUgaaeqaaOGaam4uamaaBaaaleaacaWGXbGa [email protected]@

In practice it is usually faster to compute the inverse using methods such as Gaussian elimination.

 

 

 Change of Basis.

 

Let S be a tensor, and let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@  be a Cartesian basis.  Suppose that the components of S in the basis { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@  are known to be

[ S (e) ]=[ S 11 (e) S 12 (e) S 13 (e) S 21 (e) S 22 (e) S 23 (e) S 31 (e) S 32 (e) S 33 (e) ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaCaaaleqabaGaai ikaiaahwgacaGGPaaaaOGaaiyxaiabg2da9maadmaabaqbaeqabmWa aaqaaiaadofadaqhaaWcbaGaaGymaiaaigdaaeaacaGGOaGaaCyzai aacMcaaaaakeaacaWGtbWaa0baaSqaaiaaigdacaaIYaaabaGaaiik aiaahwgacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIXaGaaG4maa qaaiaacIcacaWHLbGaaiykaaaaaOqaaiaadofadaqhaaWcbaGaaGOm aiaaigdaaeaacaGGOaGaaCyzaiaacMcaaaaakeaacaWGtbWaa0baaS qaaiaaikdacaaIYaaabaGaaiikaiaahwgacaGGPaaaaaGcbaGaam4u amaaDaaaleaacaaIYaGaaG4maaqaaiaacIcacaWHLbGaaiykaaaaaO qaaiaadofadaqhaaWcbaGaaG4maiaaigdaaeaacaGGOaGaaCyzaiaa cMcaaaaakeaacaWGtbWaa0baaSqaaiaaiodacaaIYaaabaGaaiikai aahwgacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIZaGaaG4maaqa [email protected]@

 

Now, suppose that we wish to compute the components of  S in a second Cartesian basis, { m 1 , m 2 , m 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@ .  Denote these components by

[ S (m) ]=[ S 11 (m) S 12 (m) S 13 (m) S 21 (m) S 22 (m) S 23 (m) S 31 (m) S 32 (m) S 33 (m) ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaCaaaleqabaGaai ikaiaah2gacaGGPaaaaOGaaiyxaiabg2da9maadmaabaqbaeqabmWa aaqaaiaadofadaqhaaWcbaGaaGymaiaaigdaaeaacaGGOaGaaCyBai aacMcaaaaakeaacaWGtbWaa0baaSqaaiaaigdacaaIYaaabaGaaiik aiaah2gacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIXaGaaG4maa qaaiaacIcacaWHTbGaaiykaaaaaOqaaiaadofadaqhaaWcbaGaaGOm aiaaigdaaeaacaGGOaGaaCyBaiaacMcaaaaakeaacaWGtbWaa0baaS qaaiaaikdacaaIYaaabaGaaiikaiaah2gacaGGPaaaaaGcbaGaam4u amaaDaaaleaacaaIYaGaaG4maaqaaiaacIcacaWHTbGaaiykaaaaaO qaaiaadofadaqhaaWcbaGaaG4maiaaigdaaeaacaGGOaGaaCyBaiaa cMcaaaaakeaacaWGtbWaa0baaSqaaiaaiodacaaIYaaabaGaaiikai aah2gacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIZaGaaG4maaqa [email protected]@

To do so, first compute the components of the transformation matrix [Q]

[ Q ]=[ m 1 e 1 m 1 e 2 m 1 e 3 m 2 e 1 m 2 e 2 m 2 e 3 m 3 e 1 m 3 e 2 m 3 e 3 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiaadgfaaiaawUfacaGLDb aacqGH9aqpdaWadaabaeqabaGaaCyBamaaBaaaleaacaaIXaaabeaa kiabgwSixlaahwgadaWgaaWcbaGaaGymaaqabaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWHTbWaaSbaaSqaaiaaigdaaeqa aOGaeyyXICTaaCyzamaaBaaaleaacaaIYaaabeaakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaGymaaqa baGccqGHflY1caWHLbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaaCyBam aaBaaaleaacaaIYaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaGym aaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHTb WaaSbaaSqaaiaaikdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaI YaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaah2 gadaWgaaWcbaGaaGOmaaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaa iodaaeqaaaGcbaGaaCyBamaaBaaaleaacaaIZaaabeaakiabgwSixl aahwgadaWgaaWcbaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWHTbWaaSbaaSqaaiaaiodaaeqaaOGaeyyXIC TaaCyzamaaBaaaleaacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaG4maaqabaGccqGHfl [email protected]@

(this is the same matrix you would use to transform vector components from { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  to { m 1 , m 2 , m 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  ).  Then,

[ S (m) ]=[Q][ S (e) ] [Q] T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaCaaaleqabaGaai ikaiaah2gacaGGPaaaaOGaaiyxaiabg2da9iaacUfacaWGrbGaaiyx aiaacUfacaWGtbWaaWbaaSqabeaacaGGOaGaaCyzaiaacMcaaaGcca [email protected]@

or, written out in full

[ S 11 (m) S 12 (m) S 13 (m) S 21 (m) S 22 (m) S 23 (m) S 31 (m) S 32 (m) S 33 (m) ]=[ m 1 e 1 m 1 e 2 m 1 e 3 m 2 e 1 m 2 e 2 m 2 e 3 m 3 e 1 m 3 e 2 m 3 e 3 ][ S 11 (e) S 12 (e) S 13 (e) S 21 (e) S 22 (e) S 23 (e) S 31 (e) S 32 (e) S 33 (e) ][ m 1 e 1 m 2 e 1 m 3 e 1 m 1 e 2 m 2 e 2 m 3 e 2 m 1 e 3 m 2 e 3 m 3 e 3 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb Waa0baaSqaaiaaigdacaaIXaaabaGaaiikaiaah2gacaGGPaaaaaGc baGaam4uamaaDaaaleaacaaIXaGaaGOmaaqaaiaacIcacaWHTbGaai ykaaaaaOqaaiaadofadaqhaaWcbaGaaGymaiaaiodaaeaacaGGOaGa aCyBaiaacMcaaaaakeaacaWGtbWaa0baaSqaaiaaikdacaaIXaaaba Gaaiikaiaah2gacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIYaGa aGOmaaqaaiaacIcacaWHTbGaaiykaaaaaOqaaiaadofadaqhaaWcba GaaGOmaiaaiodaaeaacaGGOaGaaCyBaiaacMcaaaaakeaacaWGtbWa a0baaSqaaiaaiodacaaIXaaabaGaaiikaiaah2gacaGGPaaaaaGcba Gaam4uamaaDaaaleaacaaIZaGaaGOmaaqaaiaacIcacaWHTbGaaiyk aaaaaOqaaiaadofadaqhaaWcbaGaaG4maiaaiodaaeaacaGGOaGaaC yBaiaacMcaaaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaqaabeqa aiaah2gadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWHLbWaaSbaaS qaaiaaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaCyBamaaBaaaleaacaaIXaaabeaakiabgwSixlaahwgadaWgaa WcbaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWHTbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBa aaleaacaaIZaaabeaaaOqaaiaah2gadaWgaaWcbaGaaGOmaaqabaGc cqGHflY1caWHLbWaaSbaaSqaaiaaigdaaeqaaOGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaaIYaaabeaa kiabgwSixlaahwgadaWgaaWcbaGaaGOmaaqabaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWHTbWaaSbaaSqaaiaaikdaaeqa aOGaeyyXICTaaCyzamaaBaaaleaacaaIZaaabeaaaOqaaiaah2gada WgaaWcbaGaaG4maaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaaigda aeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCyBam aaBaaaleaacaaIZaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaGOm aaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHTb WaaSbaaSqaaiaaiodaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaI ZaaabeaaaaGccaGLBbGaayzxaaWaamWaaeaafaqabeWadaaabaGaam 4uamaaDaaaleaacaaIXaGaaGymaaqaaiaacIcacaWHLbGaaiykaaaa aOqaaiaadofadaqhaaWcbaGaaGymaiaaikdaaeaacaGGOaGaaCyzai aacMcaaaaakeaacaWGtbWaa0baaSqaaiaaigdacaaIZaaabaGaaiik aiaahwgacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIYaGaaGymaa qaaiaacIcacaWHLbGaaiykaaaaaOqaaiaadofadaqhaaWcbaGaaGOm aiaaikdaaeaacaGGOaGaaCyzaiaacMcaaaaakeaacaWGtbWaa0baaS qaaiaaikdacaaIZaaabaGaaiikaiaahwgacaGGPaaaaaGcbaGaam4u amaaDaaaleaacaaIZaGaaGymaaqaaiaacIcacaWHLbGaaiykaaaaaO qaaiaadofadaqhaaWcbaGaaG4maiaaikdaaeaacaGGOaGaaCyzaiaa cMcaaaaakeaacaWGtbWaa0baaSqaaiaaiodacaaIZaaabaGaaiikai aahwgacaGGPaaaaaaaaOGaay5waiaaw2faamaadmaaeaqabeaacaWH TbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaaca aIXaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa h2gadaWgaaWcbaGaaGOmaaqabaGccqGHflY1caWHLbWaaSbaaSqaai aaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aCyBamaaBaaaleaacaaIZaaabeaakiabgwSixlaahwgadaWgaaWcba GaaGymaaqabaaakeaacaWHTbWaaSbaaSqaaiaaigdaaeqaaOGaeyyX ICTaaCyzamaaBaaaleaacaaIYaaabeaakiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaGOmaaqabaGccqGH flY1caWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaaIZaaabeaakiab gwSixlaahwgadaWgaaWcbaGaaGOmaaqabaaakeaacaWHTbWaaSbaaS qaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaIZaaabeaa kiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaah2gadaWgaa WcbaGaaGOmaaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaaiodaaeqa aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCyBamaaBa aaleaacaaIZaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaG4maaqa [email protected]@

 

To prove this result, let u and v be vectors satisfying

v=Su [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaC4uaiabgwSixl [email protected]@

Denote the components of u and v in the two bases by  u (e) _ , u (m) _ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadwhadaahaaWcbeqaai aacIcacaWHLbGaaiykaaaaaaGccaGGSaGaaGPaVlaaykW7caaMc8Ua aGPaVpaamaaabaGaamyDamaaCaaaleqabaGaaiikaiaah2gacaGGPa [email protected]@  and v (e) _ , v (m) _ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadAhadaahaaWcbeqaai aacIcacaWHLbGaaiykaaaaaaGccaGGSaGaaGPaVlaaykW7caaMc8Ua aGPaVpaamaaabaGaamODamaaCaaaleqabaGaaiikaiaah2gacaGGPa [email protected]@ , respectively.  Recall that the vector components are related by

u (m) _ =[Q] u (e) _ u (e) _ = [Q] T u (m) _ v (m) _ =[Q] v (e) _ v (e) _ = [Q] T v (m) _ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaamaaabaGaamyDamaaCaaale qabaGaaiikaiaah2gacaGGPaaaaaaakiabg2da9iaacUfacaWGrbGa aiyxamaamaaabaGaamyDamaaCaaaleqabaGaaiikaiaahwgacaGGPa aaaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVp aamaaabaGaamyDamaaCaaaleqabaGaaiikaiaahwgacaGGPaaaaaaa kiabg2da9iaacUfacaWGrbGaaiyxamaaCaaaleqabaGaamivaaaakm aamaaabaGaamyDamaaCaaaleqabaGaaiikaiaah2gacaGGPaaaaaaa aOqaamaamaaabaGaamODamaaCaaaleqabaGaaiikaiaah2gacaGGPa aaaaaakiabg2da9iaacUfacaWGrbGaaiyxamaamaaabaGaamODamaa CaaaleqabaGaaiikaiaahwgacaGGPaaaaaaakiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVpaamaaabaGaamODamaaCaaale qabaGaaiikaiaahwgacaGGPaaaaaaakiabg2da9iaacUfacaWGrbGa aiyxamaaCaaaleqabaGaamivaaaakmaamaaabaGaamODamaaCaaale [email protected]@

Now, we could express the tensor-vector product in either basis

v (m) _ =[ S (m) ] u (m) _ v (e) _ =[ S (e) ] u (e) _ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadAhadaahaaWcbeqaai aacIcacaWHTbGaaiykaaaaaaGccqGH9aqpdaWadaqaaiaadofadaah aaWcbeqaaiaacIcacaWHTbGaaiykaaaaaOGaay5waiaaw2faamaama aabaGaamyDamaaCaaaleqabaGaaiikaiaah2gacaGGPaaaaaaakiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaWaaaeaaca WG2bWaaWbaaSqabeaacaGGOaGaaCyzaiaacMcaaaaaaOGaeyypa0Za amWaaeaacaWGtbWaaWbaaSqabeaacaGGOaGaaCyzaiaacMcaaaaaki aawUfacaGLDbaadaadaaqaaiaadwhadaahaaWcbeqaaiaacIcacaWH [email protected]@

Substitute for u (e) _ , v (e) _ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadwhadaahaaWcbeqaai aacIcacaWHLbGaaiykaaaaaaGccaGGSaGaaGPaVlaaykW7caaMc8Ua aGPaVpaamaaabaGaamODamaaCaaaleqabaGaaiikaiaahwgacaGGPa [email protected]@  from above into the second of these two relations, we see that

[ Q ] T v (m) _ =[ S (e) ] [ Q ] T u (m) _ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiaadgfaaiaawUfacaGLDb aadaahaaWcbeqaaiaadsfaaaGcdaadaaqaaiaadAhadaahaaWcbeqa aiaacIcacaWHTbGaaiykaaaaaaGccqGH9aqpdaWadaqaaiaadofada ahaaWcbeqaaiaacIcacaWHLbGaaiykaaaaaOGaay5waiaaw2faamaa dmaabaGaamyuaaGaay5waiaaw2faamaaCaaaleqabaGaamivaaaakm aamaaabaGaamyDamaaCaaaleqabaGaaiikaiaah2gacaGGPaaaaaaa [email protected]@

Recall that

[ Q ] [ Q ] T =[ I ][ I ] v (m) _ = v (m) _ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiaadgfaaiaawUfacaGLDb aadaWadaqaaiaadgfaaiaawUfacaGLDbaadaahaaWcbeqaaiaadsfa aaGccqGH9aqpdaWadaqaaiaadMeaaiaawUfacaGLDbaacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8+aamWaaeaacaWGjbaacaGLBbGaayzxaaWaaWaaaeaacaWG 2bWaaWbaaSqabeaacaGGOaGaaCyBaiaacMcaaaaaaOGaeyypa0ZaaW aaaeaacaWG2bWaaWbaaSqabeaacaGGOaGaaCyBaiaacMcaaaaaaaaa @[email protected]

so multiplying both sides by [Q] shows that

v (m) _ =[ Q ][ S (e) ] [ Q ] T u (m) _ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadAhadaahaaWcbeqaai aacIcacaWHTbGaaiykaaaaaaGccqGH9aqpdaWadaqaaiaadgfaaiaa wUfacaGLDbaadaWadaqaaiaadofadaahaaWcbeqaaiaacIcacaWHLb GaaiykaaaaaOGaay5waiaaw2faamaadmaabaGaamyuaaGaay5waiaa w2faamaaCaaaleqabaGaamivaaaakmaamaaabaGaamyDamaaCaaale [email protected]@

so, comparing with the first of equation (1)

[ S (m) ]=[Q][ S (e) ] [Q] T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaCaaaleqabaGaai ikaiaah2gacaGGPaaaaOGaaiyxaiabg2da9iaacUfacaWGrbGaaiyx aiaacUfacaWGtbWaaWbaaSqabeaacaGGOaGaaCyzaiaacMcaaaGcca [email protected]@

as stated.

 

In index notation, we would write

S ij (m) = Q ik S kl (e) Q jl Q ij = m i e j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaa0baaS qaaiaadMgacaWGQbaabaGaaiikaiaah2gacaGGPaaaaOGaeyypa0Ja amyuamaaBaaaleaacaWGPbGaam4AaaqabaGccaWGtbWaa0baaSqaai aadUgacaWGSbaabaGaaiikaiaahwgacaGGPaaaaOGaamyuamaaBaaa leaacaWGQbGaamiBaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWGrbWaaSbaaSqaaiaadMgacaWGQbaabeaaki abg2da9iaah2gadaWgaaWcbaGaamyAaaqabaGccqGHflY1caWHLbWa [email protected]@

Another, perhaps cleaner, way to derive this result is to expand the two tensors as the appropriate dyadic products of the basis vectors

S kl (m) m k m l = S kl (e) e k e l m i [ S kl (m) m k m l ] m j = m i S kl (e) e k e l m j S ij (m) =( m i e k ) S kl (e) ( e l m j ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadofada qhaaWcbaGaam4AaiaadYgaaeaacaGGOaGaaCyBaiaacMcaaaGccaWH TbWaaSbaaSqaaiaadUgaaeqaaOGaey4LIqSaaCyBamaaBaaaleaaca WGSbaabeaakiabg2da9iaadofadaqhaaWcbaGaam4AaiaadYgaaeaa caGGOaGaaCyzaiaacMcaaaGccaWHLbWaaSbaaSqaaiaadUgaaeqaaO Gaey4LIqSaaCyzamaaBaaaleaacaWGSbaabeaaaOqaaiabgkDiElaa h2gadaWgaaWcbaGaamyAaaqabaGccqGHflY1daWadaqaaiaadofada qhaaWcbaGaam4AaiaadYgaaeaacaGGOaGaaCyBaiaacMcaaaGccaWH TbWaaSbaaSqaaiaadUgaaeqaaOGaey4LIqSaaCyBamaaBaaaleaaca WGSbaabeaaaOGaay5waiaaw2faaiabgwSixlaah2gadaWgaaWcbaGa amOAaaqabaGccqGH9aqpcaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaey yXICTaam4uamaaDaaaleaacaWGRbGaamiBaaqaaiaacIcacaWHLbGa aiykaaaakiaahwgadaWgaaWcbaGaam4AaaqabaGccqGHxkcXcaWHLb WaaSbaaSqaaiaadYgaaeqaaOGaeyyXICTaaCyBamaaBaaaleaacaWG QbaabeaaaOqaaiabgkDiElaadofadaqhaaWcbaGaamyAaiaadQgaae aacaGGOaGaaCyBaiaacMcaaaGccqGH9aqpcaGGOaGaaCyBamaaBaaa leaacaWGPbaabeaakiabgwSixlaahwgadaWgaaWcbaGaam4Aaaqaba GccaGGPaGaam4uamaaDaaaleaacaWGRbGaamiBaaqaaiaacIcacaWH LbGaaiykaaaakiaacIcacaWHLbWaaSbaaSqaaiaadYgaaeqaaOGaey [email protected]@

 

 

 

 Invariants

 

Invariants of a tensor are scalar functions of the tensor components which remain constant under a basis change.  That is to say, the invariant has the same value when computed in two arbitrary bases { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  and { m 1 , m 2 , m 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@ .  A symmetric second order tensor always has three independent invariants.

 

Examples of invariants are

1.      The three eigenvalues

2.      The determinant

3.      The trace

4.      The inner and outer products

 

These are not all independent [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  for example any of 2-4 can be calculated in terms of 1.

 

 

 

 

In practice, the most commonly used invariants are:

I 1 =trace(S)= S kk I 2 = 1 2 (trace (S) 2 SS)= 1 2 ( S ii S jj S ij S ji ) I 3 =det(S)= 1 6 ijk pqr S ip S jq S kr = ijk S i1 S j2 S k3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadMeada WgaaWcbaGaaGymaaqabaGccqGH9aqpciGG0bGaaiOCaiaacggacaGG JbGaaiyzaiGacIcacaWHtbGaaiykaiabg2da9iaadofadaWgaaWcba Gaam4AaiaadUgaaeqaaaGcbaGaamysamaaBaaaleaacaaIYaaabeaa kiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikaiGacshaca GGYbGaaiyyaiaacogacaGGLbGaaiikaiaahofacaGGPaWaaWbaaSqa beaacaaIYaaaaOGaeyOeI0IaaC4uaiabgwSixlabgwSixlaahofaca GGPaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaam4u amaaBaaaleaacaWGPbGaamyAaaqabaGccaWGtbWaaSbaaSqaaiaadQ gacaWGQbaabeaakiabgkHiTiaadofadaWgaaWcbaGaamyAaiaadQga aeqaaOGaam4uamaaBaaaleaacaWGQbGaamyAaaqabaGccaGGPaaaba GaamysamaaBaaaleaacaaIZaaabeaakiabg2da9iGacsgacaGGLbGa aiiDaiaacIcacaWHtbGaaiykaiabg2da9maalaaabaGaaGymaaqaai aaiAdaaaGaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGc cqGHiiIZdaWgaaWcbaGaamiCaiaadghacaWGYbaabeaakiaadofada WgaaWcbaGaamyAaiaadchaaeqaaOGaam4uamaaBaaaleaacaWGQbGa amyCaaqabaGccaWGtbWaaSbaaSqaaiaadUgacaWGYbaabeaakiabg2 da9iabgIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaam4u amaaBaaaleaacaWGPbGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaadQ gacaaIYaaabeaakiaadofadaWgaaWcbaGaam4Aaiaaiodaaeqaaaaa [email protected]@

 

 

 Eigenvalues and Eigenvectors (Principal values and direction)

 

Let S be a second order tensor.  The scalars λ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  and unit vectors m which satisfy

Sm=λm [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyXICTaaCyBaiabg2da9i [email protected]@

are known as the eigenvalues and eigenvectors of S, or the principal values and principal directions of S. Note that λ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  may be complex.  For a second order tensor in three dimensions, there are generally three values of λ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  and three unique unit vectors m which satisfy this equation.  Occasionally, there may be only two or one value of λ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ .  If this is the case, there are infinitely many possible vectors m that satisfy the equation.  The eigenvalues of a tensor, and the components of the eigenvectors, may be computed by finding the eigenvalues and eigenvectors of the matrix of components.

 

The eigenvalues of a symmetric tensor are always real, and its eigenvectors are mutually perpendicular (these two results are important and are proved below).  The eigenvalues of a skew tensor are always pure imaginary or zero.

 

The eigenvalues of a second order tensor are computed using the condition det(SλI)=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGKbGaaiyzai aacshacaGGOaGaaC4uaiabgkHiTiabeU7aSjaahMeacaGGPaGaeyyp [email protected]@ .  This yields a cubic equation, which can be expressed as

λ 3 I 1 λ 2 + I 2 λ I 3 =0 I 1 =trace(S), I 2 =( I 1 2 SS)/2 I 3 =det(S) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaa WcbeqaaiaaiodaaaGccqGHsislcaWGjbWaaSbaaSqaaiaaigdaaeqa aOGaeq4UdW2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamysamaaBa aaleaacaaIYaaabeaakiabeU7aSjabgkHiTiaadMeadaWgaaWcbaGa aG4maaqabaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGjbWa aSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamiDaiaadkhacaWGHbGaam 4yaiaadwgacaGGOaGaaC4uaiaacMcacaGGSaGaaGzaVlaaykW7caaM c8UaaGPaVlaaykW7caWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0 JaaiikaiaadMeadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHsisl caWHtbGaeyyXICTaeyyXICTaaC4uaiaacMcacaGGVaGaaGOmaiaayk W7caaMc8UaaGPaVlaaykW7caWGjbWaaSbaaSqaaiaaiodaaeqaaOGa [email protected]@

There are various ways to solve the resulting cubic equation explicitly [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  a solution for symmetric S is given below, but the results for a general tensor are too messy to be given here.   The eigenvectors are then computed from the condition (SλI)m=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGOaGaaC4uai [email protected]@ .

 

 The Cayley-Hamilton Theorem

 

Let S be a second order tensor and let I 1 =trace(S), I 2 =( I 1 2 SS)/2 I 3 =det(S) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaSbaaS qaaiaaigdaaeqaaOGaeyypa0JaamiDaiaadkhacaWGHbGaam4yaiaa dwgacaGGOaGaaC4uaiaacMcacaGGSaGaaGzaVlaaykW7caaMc8UaaG PaVlaaykW7caWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaaiik aiaadMeadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHsislcaWHtb GaeyyXICTaeyyXICTaaC4uaiaacMcacaGGVaGaaGOmaiaaykW7caaM c8UaaGPaVlaaykW7caWGjbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0 [email protected]@  be the three invariants.   Then

S 3 I 1 S 2 + I 2 S I 3 =0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbWaaWbaaS qabeaacaaIZaaaaOGaeyOeI0IaamysamaaBaaaleaacaaIXaaabeaa kiaahofadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGjbWaaSbaaS qaaiaaikdaaeqaaOGaaC4uaiabgkHiTiaadMeadaWgaaWcbaGaaG4m [email protected]@

(i.e. a tensor satisfies its characteristic equation).   There is an obscure trick to show this… Consider the tensor SαI [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbGaeyOeI0 [email protected]@  (where α [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 [email protected]@  is an arbitrary scalar), and let T be the adjoint of SαI [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbGaeyOeI0 [email protected]@ , (the adjoint is just the inverse multiplied by the determinant) which satisfies

T(SαI)=det(SαI)I=( α 3 + I 1 α 2 I 2 α+ I 3 )I [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHubGaaiikai aahofacqGHsislcqaHXoqycaWHjbGaaiykaiabg2da9iGacsgacaGG LbGaaiiDaiaacIcacaWHtbGaeyOeI0IaeqySdeMaaCysaiaacMcaca WHjbGaeyypa0JaaiikaiabgkHiTiabeg7aHnaaCaaaleqabaGaaG4m aaaakiabgUcaRiaadMeadaWgaaWcbaGaaGymaaqabaGccqaHXoqyda ahaaWcbeqaaiaaikdaaaGccqGHsislcaWGjbWaaSbaaSqaaiaaikda aeqaaOGaeqySdeMaey4kaSIaamysamaaBaaaleaacaaIZaaabeaaki [email protected]@

Assume that T= α 2 T 1 +α T 2 + T 3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqydaahaa WcbeqaaiaaikdaaaGccaWHubWaaSbaaSqaaiaaigdaaeqaaOGaey4k aSIaeqySdeMaaCivamaaBaaaleaacaaIYaaabeaakiabgUcaRiaahs [email protected]@ .   Substituting in the preceding equation shows that

T 1 =I T 1 S T 2 = I 1 I T 3 T 2 S= I 2 I T 3 S= I 3 I [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHubWaaSbaaS qaaiaaigdaaeqaaOGaeyypa0JaaCysaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaahsfadaWgaaWcba GaaGymaaqabaGccaWHtbGaeyOeI0IaaCivamaaBaaaleaacaaIYaaa beaakiabg2da9iaadMeadaWgaaWcbaGaaGymaaqabaGccaWHjbGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaCivamaaBaaaleaacaaIZaaabeaaki abgkHiTiaahsfadaWgaaWcbaGaaGOmaaqabaGccaWHtbGaeyypa0Ja amysamaaBaaaleaacaaIYaaabeaakiaahMeacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caWHubWaaSbaaSqaaiaaiodaae qaaOGaaC4uaiabg2da9iaadMeadaWgaaWcbaGaaG4maaqabaGccaWH [email protected]@

Use these to substitute for I 1 , I 2 , I 3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaSbaaS qaaiaaigdaaeqaaOGaaiilaiaadMeadaWgaaWcbaGaaGOmaaqabaGc [email protected]@  into

S 3 I 1 S 2 + I 2 S I 3 = S 3 (S T 2 ) S 2 +( T 3 T 2 S)S T 3 S=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbWaaWbaaS qabeaacaaIZaaaaOGaeyOeI0IaamysamaaBaaaleaacaaIXaaabeaa kiaahofadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGjbWaaSbaaS qaaiaaikdaaeqaaOGaaC4uaiabgkHiTiaadMeadaWgaaWcbaGaaG4m aaqabaGccqGH9aqpcaWHtbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0 IaaiikaiaahofacqGHsislcaWHubWaaSbaaSqaaiaaikdaaeqaaOGa aiykaiaahofadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGaaC ivamaaBaaaleaacaaIZaaabeaakiabgkHiTiaahsfadaWgaaWcbaGa aGOmaaqabaGccaWHtbGaaiykaiaahofacqGHsislcaWHubWaaSbaaS [email protected]@

 

 

 

 

3 SPECIAL TENSORS

 

 Identity tensor  The identity tensor I is the tensor such that, for any tensor S or vector v

Iv=vI=v SI=IS=S [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahMeacqGHflY1caWH2bGaey ypa0JaaCODaiabgwSixlaahMeacqGH9aqpcaWH2baabaGaaC4uaiab gwSixlaahMeacqGH9aqpcaWHjbGaeyyXICTaaC4uaiabg2da9iaaho [email protected]@

In any basis, the identity tensor has components

[ 1 0 0 0 1 0 0 0 1 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaaqaaiaaicda [email protected]@

 

 Symmetric Tensor A symmetric tensor S has the property

S= S T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyypa0JaaC4uamaaCaaale [email protected]@

The components of a symmetric tensor have the form

[ S 11 S 12 S 13 S 12 S 22 S 23 S 13 S 23 S 33 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb WaaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadofadaWgaaWcbaGa aGymaiaaikdaaeqaaaGcbaGaam4uamaaBaaaleaacaaIXaGaaG4maa qabaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaa dofadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaam4uamaaBaaale aacaaIYaGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaI ZaaabeaaaOqaaiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcba Gaam4uamaaBaaaleaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzx [email protected]@

so that there are only six independent components of the tensor, instead of nine.   Symmetric tensors have some nice properties:

 

·         The eigenvectors of a symmetric tensor with distinct eigenvalues are orthogonal.   To see this, let u,v [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWH1bGaaiilai [email protected]@  be two eigenvectors, with corresponding eigenvalues λ u , λ v [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaWgaa WcbaGaamyDaaqabaGccaGGSaGaeq4UdW2aaSbaaSqaaiaadAhaaeqa [email protected]@ . Then v[ Su ]=u[ S T v ]=u[ Sv ]v λ u u=u λ v v( λ u λ v )uv=0uv=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWH2bGaeyyXIC 9aamWaaeaacaWHtbGaeyyXICTaaCyDaaGaay5waiaaw2faaiabg2da 9iaahwhacqGHflY1daWadaqaaiaahofadaahaaWcbeqaaiaadsfaaa GccqGHflY1caWH2baacaGLBbGaayzxaaGaeyypa0JaaCyDaiabgwSi xpaadmaabaGaaC4uaiabgwSixlaahAhaaiaawUfacaGLDbaacqGHsh I3caWH2bGaeyyXICTaeq4UdW2aaSbaaSqaaiaadwhaaeqaaOGaaCyD aiabg2da9iaahwhacqGHflY1cqaH7oaBdaWgaaWcbaGaamODaaqaba GccaWH2bGaeyO0H4TaaiikaiabeU7aSnaaBaaaleaacaWG1baabeaa kiabgkHiTiabeU7aSnaaBaaaleaacaWG2baabeaakiaacMcacaWH1b GaeyyXICTaaCODaiabg2da9iaaicdacqGHshI3caWH1bGaeyyXICTa [email protected]@ .

 

·         The eigenvalues of a symmetric tensor are realTo see this, suppose that λ,u [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBcaGGSa [email protected]@  are a complex eigenvalue/eigenvector pair, and let λ ¯ , u ¯ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBgaqeai [email protected]@  denote their complex conjugates.   Then, by definition Su=λuS u ¯ = λ ¯ u ¯ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbGaeyyXIC TaaCyDaiabg2da9iabeU7aSjaahwhacqGHshI3caWHtbGabCyDayaa [email protected]@ .  And hence u ¯ [ Su ]=λ u ¯ u,u[ S u ¯ ]= λ ¯ u u ¯ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWH1bGbaebacq GHflY1daWadaqaaiaahofacqGHflY1caWH1baacaGLBbGaayzxaaGa eyypa0Jaeq4UdWMabCyDayaaraGaeyyXICTaaCyDaiaacYcacaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWH1bGaeyyXIC9aamWa aeaacaWHtbGaeyyXICTabCyDayaaraaacaGLBbGaayzxaaGaeyypa0 [email protected]@ .  But note that for a symmetric tensor u ¯ [ Su ]=u[ S T u ¯ ]=u[ S u ¯ ] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWH1bGbaebacq GHflY1daWadaqaaiaahofacqGHflY1caWH1baacaGLBbGaayzxaaGa eyypa0JaaCyDaiabgwSixpaadmaabaGaaC4uamaaCaaaleqabaGaam ivaaaakiabgwSixlqahwhagaqeaaGaay5waiaaw2faaiabg2da9iaa hwhacqGHflY1daWadaqaaiaahofacqGHflY1ceWH1bGbaebaaiaawU [email protected]@ .  Thus λ u ¯ u= λ ¯ u u ¯ λ= λ ¯ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBceWH1b GbaebacqGHflY1caWH1bGaeyypa0Jafq4UdWMbaebacaWH1bGaeyyX ICTabCyDayaaraGaeyO0H4Taeq4UdWMaeyypa0Jafq4UdWMbaebaaa [email protected]@ .

 

The eigenvalues of a symmetric tensor can be computed as

λ k = I 1 3 +2 p 3 cos{ 1 3 cos 1 ( 3q 2p 3 p ) 2(k1)π 3 }k=1,2,3 p= I 2 1 3 I 1 2 q= 2 I 1 3 9 I 1 I 2 +27 I 3 27 I 1 =trace(S) I 2 = 1 2 ( I 1 2 S:S ) I 3 =det(S) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiabeU7aSn aaBaaaleaacaWGRbaabeaakiabg2da9maalaaabaGaamysamaaBaaa leaacaaIXaaabeaaaOqaaiaaiodaaaGaey4kaSIaaGOmamaakaaaba WaaSaaaeaacqGHsislcaWGWbaabaGaaG4maaaaaSqabaGcciGGJbGa ai4BaiaacohadaGadaqaamaalaaabaGaaGymaaqaaiaaiodaaaGaci 4yaiaac+gacaGGZbWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaeWa aeaadaWcaaqaaiaaiodacaWGXbaabaGaaGOmaiaadchaaaWaaOaaae aadaWcaaqaaiabgkHiTiaaiodaaeaacaWGWbaaaaWcbeaaaOGaayjk aiaawMcaaiabgkHiTmaalaaabaGaaGOmaiaacIcacaWGRbGaeyOeI0 IaaGymaiaacMcacqaHapaCaeaacaaIZaaaaaGaay5Eaiaaw2haaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caWGRbGaeyypa0JaaGymaiaa cYcacaaIYaGaaiilaiaaiodaaeaacaWGWbGaeyypa0JaamysamaaBa aaleaacaaIYaaabeaakiabgkHiTmaalaaabaGaaGymaaqaaiaaioda aaGaamysamaaDaaaleaacaaIXaaabaGaaGOmaaaakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caWGXbGaeyypa0JaeyOeI0YaaSaaaeaacaaIYaGaamysamaaDaaale aacaaIXaaabaGaaG4maaaakiabgkHiTiaaiMdacaWGjbWaaSbaaSqa aiaaigdaaeqaaOGaamysamaaBaaaleaacaaIYaaabeaakiabgUcaRi aaikdacaaI3aGaamysamaaBaaaleaacaaIZaaabeaaaOqaaiaaikda caaI3aaaaaqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccqGH9aqpca WG0bGaamOCaiaadggacaWGJbGaamyzaiaacIcacaWHtbGaaiykaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaa aeaacaaIXaaabaGaaGOmaaaadaqadaqaaiaadMeadaqhaaWcbaGaaG ymaaqaaiaaikdaaaGccqGHsislcaWHtbGaaiOoaiaahofaaiaawIca caGLPaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caWGjbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Ja [email protected]@

The eigenvectors can then be found by back-substitution into [ SλI ]m=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaaiaaho facqGHsislcqaH7oaBcaWHjbaacaGLBbGaayzxaaGaeyyXICTaaCyB [email protected]@ . To do this, note that the matrix equation can be written as

[ S 11 λ S 12 S 13 S 12 S 22 λ S 23 S 13 S 23 S 33 λ ][ m 1 m 2 m 3 ]=[ 0 0 0 ] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaauaabe qadmaaaeaacaWGtbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgkHi TiabeU7aSbqaaiaadofadaWgaaWcbaGaaGymaiaaikdaaeqaaaGcba Gaam4uamaaBaaaleaacaaIXaGaaG4maaqabaaakeaacaWGtbWaaSba aSqaaiaaigdacaaIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaGOmai aaikdaaeqaaOGaeyOeI0Iaeq4UdWgabaGaam4uamaaBaaaleaacaaI YaGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIZaaabe aaaOqaaiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaam4u amaaBaaaleaacaaIZaGaaG4maaqabaGccqGHsislcqaH7oaBaaaaca GLBbGaayzxaaWaamWaaeaafaqabeWabaaabaGaamyBamaaBaaaleaa caaIXaaabeaaaOqaaiaad2gadaWgaaWcbaGaaGOmaaqabaaakeaaca WGTbWaaSbaaSqaaiaaiodaaeqaaaaaaOGaay5waiaaw2faaiabg2da 9maadmaabaqbaeqabmqaaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa [email protected]@

Since the determinant of the matrix is zero, we can discard any row in the equation system and take any column over to the right hand side.   For example, if the tensor has at least one eigenvector with m 3 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaSbaaS [email protected]@  then the values of m 1 , m 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaSbaaS qaaiaaigdaaeqaaOGaaiilaiaad2gadaWgaaWcbaGaaGOmaaqabaaa [email protected]@  for this eigenvector can be found by discarding the third row, and writing

[ S 11 λ S 12 S 12 S 22 λ ][ m 1 m 2 ]= m 3 [ S 13 S 23 ] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaauaabe qaciaaaeaacaWGtbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgkHi TiabeU7aSbqaaiaadofadaWgaaWcbaGaaGymaiaaikdaaeqaaaGcba Gaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWaaSba aSqaaiaaikdacaaIYaaabeaakiabgkHiTiabeU7aSbaaaiaawUfaca GLDbaadaWadaqaauaabeqaceaaaeaacaWGTbWaaSbaaSqaaiaaigda aeqaaaGcbaGaamyBamaaBaaaleaacaaIYaaabeaaaaaakiaawUfaca GLDbaacqGH9aqpcqGHsislcaWGTbWaaSbaaSqaaiaaiodaaeqaaOWa amWaaeaafaqabeGabaaabaGaam4uamaaBaaaleaacaaIXaGaaG4maa qabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaaaaaakiaa [email protected]@

 

·         Spectral decomposition of a symmetric tensor  Let S be a symmetric second order tensor, and let { λ i , e i } [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaeq4UdW 2aaSbaaSqaaiaadMgaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaamyA [email protected]@  be the three eigenvalues and eigenvectors of S.  Then S can be expressed as

S= i=1 3 λ i e i e i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbGaeyypa0 ZaaabCaeaacqaH7oaBdaWgaaWcbaGaamyAaaqabaGccaWHLbWaaSba aSqaaiaadMgaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacaWGPbaabe [email protected]@

To see this, note that S can always be expanded as a sum of 9 dyadic products of an orthogonal basis.  S= S ij e i e j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbGaeyypa0 Jaam4uamaaBaaaleaacaWGPbGaamOAaaqabaGccaWHLbWaaSbaaSqa aiaadMgaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacaWGQbaabeaaaa [email protected]@ But since e i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHLbWaaSbaaS [email protected]@  are eigenvectors it follows that e m ( S ij e i e j ) e k = S mk ={ λ k m=k 0mk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHLbWaaSbaaS qaaiaad2gaaeqaaOGaeyyXICTaaiikaiaadofadaWgaaWcbaGaamyA aiaadQgaaeqaaOGaaCyzamaaBaaaleaacaWGPbaabeaakiabgEPiel aahwgadaWgaaWcbaGaamOAaaqabaGccaGGPaGaeyyXICTaaCyzamaa BaaaleaacaWGRbaabeaakiabg2da9iaadofadaWgaaWcbaGaamyBai aadUgaaeqaaOGaeyypa0ZaaiqaaeaafaqabeGabaaabaGaeq4UdW2a aSbaaSqaaiaadUgaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaad2 gacqGH9aqpcaWGRbaabaGaaGimaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWGTbGaeyiyIKRaam4AaaaaaiaawUhaaa [email protected]@

 

 

 Skew Tensor.  A skew tensor S has the property

S T =S [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbWaaWbaaSqabeaacaWGubaaaO G[email protected]@

The components of a skew tensor have the form

[ 0 S 12 S 13 S 12 0 S 23 S 13 S 23 0 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaaIWa aabaGaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWa aSbaaSqaaiaaigdacaaIZaaabeaaaOqaaiabgkHiTiaadofadaWgaa WcbaGaaGymaiaaikdaaeqaaaGcbaGaaGimaaqaaiaadofadaWgaaWc baGaaGOmaiaaiodaaeqaaaGcbaGaeyOeI0Iaam4uamaaBaaaleaaca aIXaGaaG4maaqabaaakeaacqGHsislcaWGtbWaaSbaaSqaaiaaikda [email protected]@

 

Every second-order skew tensor has a dual vector w that satisfies

Su=w×u [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHtbGaeyyXIC [email protected]@

for all vectors u.   You can see this by noting that S 12 = w 3 S 13 = w 2 S 23 = w 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaaigdacaaIYaaabeaakiabg2da9iabgkHiTiaadEhadaWgaaWc baGaaG4maaqabaGccaaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaBa aaleaacaaIXaGaaG4maaqabaGccqGH9aqpcaWG3bWaaSbaaSqaaiaa ikdaaeqaaOGaaGPaVlaaykW7caaMc8Uaam4uamaaBaaaleaacaaIYa GaaG4maaqabaGccqGH9aqpcqGHsislcaWG3bWaaSbaaSqaaiaaigda [email protected]@  and expanding out the tensor and cross products explicitly.  In index notation, we can also write

S ij = ijk w k w i = 1 2 ijk S jk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaadMgacaWGQbaabeaakiabg2da9iabgkHiTiabgIGiopaaBaaa leaacaWGPbGaamOAaiaadUgaaeqaaOGaam4DamaaBaaaleaacaWGRb aabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4DamaaBaaaleaacaWG Pbaabeaakiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaa GaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccaWGtbWa [email protected]@ .

 

 Orthogonal Tensors An orthogonal tensor R has the property

R R T = R T R=I R 1 = R T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahkfacqGHflY1caWHsbWaaW baaSqabeaacaWGubaaaOGaeyypa0JaaCOuamaaCaaaleqabaGaamiv aaaakiabgwSixlaahkfacqGH9aqpcaWHjbaabaGaaCOuamaaCaaale qabaGaeyOeI0IaaGymaaaakiabg2da9iaahkfadaahaaWcbeqaaiaa [email protected]@

An orthogonal tensor must have det(S)=±1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC [email protected]@ ; a tensor with det(S)=+1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC [email protected]@  is known as a proper orthogonal tensor.  Orthogonal tensors also have some interesting and useful properties:

·         Orthogonal tensors map a vector onto another vector with the same length.  To see this, let u be an arbitrary vector.  Then, note that | Ru | 2 =[ Ru ][ Ru ]=u R T Ru=uu= | u | 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaabdaqaaiaahk facqGHflY1caWH1baacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaa aOGaeyypa0ZaamWaaeaacaWHsbGaeyyXICTaaCyDaaGaay5waiaaw2 faaiabgwSixpaadmaabaGaaCOuaiabgwSixlaahwhaaiaawUfacaGL DbaacqGH9aqpcaWH1bGaeyyXICTaaCOuamaaCaaaleqabaGaamivaa aakiaahkfacqGHflY1caWH1bGaeyypa0JaaCyDaiabgwSixlaahwha cqGH9aqpdaabdaqaaiaahwhaaiaawEa7caGLiWoadaahaaWcbeqaai [email protected]@

·         The eigenvalues of an orthogonal tensor are 1, e ±iθ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaiilai [email protected]@  for some value of θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 [email protected]@ .  To see this, let u [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 [email protected]@  be an eigenvector, with corresponding eigenvalue λ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 [email protected]@ .  By definition, Ru=λu [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHsbGaeyyXIC [email protected]@ .  Hence, [ Ru ][ Ru ]=λuλuuu= λ 2 uu λ 2 =1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaaiaahk facqGHflY1caWH1baacaGLBbGaayzxaaGaeyyXIC9aamWaaeaacaWH sbGaeyyXICTaaCyDaaGaay5waiaaw2faaiabg2da9iabeU7aSjaahw hacqGHflY1cqaH7oaBcaWH1bGaeyO0H4TaaCyDaiabgwSixlaahwha cqGH9aqpcqaH7oaBdaahaaWcbeqaaiaaikdaaaGccaWH1bGaeyyXIC TaaCyDaiabgkDiElabeU7aSnaaCaaaleqabaGaaGOmaaaakiabg2da [email protected]@ . Similarly, λ λ ¯ =1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBcuaH7o [email protected]@ .  Since the characteristic equation is cubic, there must be at most three eigenvalues, and at least one eigenvalue must be real.

 

Proper orthogonal tensors can be visualized physically as rotations.  A rotation can also be represented in several other forms besides a proper orthogonal tensor.   For example

·         The Rodriguez representation quantifies a rotation as an angle of rotation θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 [email protected]@  (in radians) about some axis n (specified by a unit vector).  Given R, there are various ways to compute n  and θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 [email protected]@ .  For example, one way would be find the eigenvalues and the real eigenvector.  The real eigenvector (suitably normalized) must correspond to n; the complex eigenvalues give e iθ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGLbWaaWbaaS [email protected]@ .  A faster method is to note that

trace(R)=1+2cosθ2sinθn=dual(R R T ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqG0bGaaeOCai aabggacaqGJbGaaeyzaiaacIcacaWHsbGaaiykaiabg2da9iaaigda cqGHRaWkcaaIYaGaci4yaiaac+gacaGGZbGaeqiUdeNaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaikdaciGGZbGaaiyA aiaac6gacqaH4oqCcaWHUbGaeyypa0JaaeizaiaabwhacaqGHbGaae iBaiaacIcacaWHsbGaeyOeI0IaaCOuamaaCaaaleqabaGaamivaaaa [email protected]@

·         Alternatively, given n and θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 [email protected]@ , R can be computed from

R=cosθI+WW(1cosθ)+Wsinθ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHsbGaeyypa0 Jaci4yaiaac+gacaGGZbGaeqiUdeNaaCysaiabgUcaRiaahEfacqGH flY1caWHxbGaaiikaiaaigdacqGHsislciGGJbGaai4Baiaacohacq aH4oqCcaGGPaGaey4kaSIaaC4vaiGacohacaGGPbGaaiOBaiabeI7a [email protected]@

where W is the skew tensor that has n as its dual vector, i.e. W ij = ijk n k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaSbaaS qaaiaadMgacaWGQbaabeaakiabg2da9iabgkHiTiabgIGiopaaBaaa leaacaWGPbGaamOAaiaadUgaaeqaaOGaamOBamaaBaaaleaacaWGRb [email protected]@ .  In index notation, this formula is

R ij =cosθ δ ij + n i n j (1cosθ)sinθ ijk n k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaSbaaS qaaiaadMgacaWGQbaabeaakiabg2da9iGacogacaGGVbGaai4Caiab eI7aXjabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccqGHRaWkca WGUbWaaSbaaSqaaiaadMgaaeqaaOGaamOBamaaBaaaleaacaWGQbaa beaakiaacIcacaaIXaGaeyOeI0Iaci4yaiaac+gacaGGZbGaeqiUde NaaiykaiabgkHiTiGacohacaGGPbGaaiOBaiabeI7aXjabgIGiopaa BaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamOBamaaBaaaleaaca [email protected]@

 

 

 

Another useful result is the Polar Decomposition Theorem, which states that invertible second order tensors can be expressed as a product of a symmetric tensor with an orthogonal tensor: 

A=RU=VRR R T =IU= U T V= V T [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbGaeyypa0 JaaCOuaiabgwSixlaahwfacqGH9aqpcaWHwbGaeyyXICTaaCOuaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCOuaiaahk fadaahaaWcbeqaaiaadsfaaaGccqGH9aqpcaWHjbGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWHvbGaeyypa0JaaCyvamaaCaaaleqabaGaamivaaaakiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCOvaiabg2da9iaahAfada [email protected]@

Moreover, the tensors R,U,V [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHsbGaaiilai [email protected]@  are unique.  To see this, note that

·         A T A [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbWaaWbaaS [email protected]@  is symmetric and has positive eigenvalues (to see that it’s symmetric, simply take the transpose, and to see that the eigenvalues are positive, note that dx( A T A)dx>0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaaCiEai abgwSixlaacIcacaWHbbWaaWbaaSqabeaacaWGubaaaOGaeyyXICTa [email protected]@  for all vectors dx). 

·         Let  λ k 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaqhaa [email protected]@  and m k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHTbWaaSbaaS [email protected]@  be the three eigenvalues and eigenvectors of A T A [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbWaaWbaaS [email protected]@ .  Since the eigenvectors are orthogonal, we can write A T A= k=1 3 λ k 2 m k m k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbWaaWbaaS qabeaacaWGubaaaOGaaCyqaiabg2da9maaqahabaGaeq4UdW2aa0ba aSqaaiaadUgaaeaacaaIYaaaaOGaaCyBamaaBaaaleaacaWGRbaabe aakiabgEPielaah2gadaWgaaWcbaGaam4AaaqabaaabaGaam4Aaiab [email protected]@

·         We can then set U= k=1 3 λ k m k m k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHvbGaeyypa0 ZaaabCaeaacqaH7oaBdaWgaaWcbaGaam4AaaqabaGccaWHTbWaaSba aSqaaiaadUgaaeqaaOGaey4LIqSaaCyBamaaBaaaleaacaWGRbaabe [email protected]@  and define R=A U 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHsbGaeyypa0 [email protected]@ U is clearly symmetric, and also U 2 = A T A [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHvbWaaWbaaS qabeaacaaIYaaaaOGaeyypa0JaaCyqamaaCaaaleqabaGaamivaaaa [email protected]@ . To see that R is orthogonal note that: R T R= U T A T A U 1 = U T U 2 U 1 =I [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHsbWaaWbaaS qabeaacaWGubaaaOGaaCOuaiabg2da9iaahwfadaahaaWcbeqaaiab gkHiTiaadsfaaaGccaWHbbWaaWbaaSqabeaacaWGubaaaOGaaCyqai aahwfadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpcaWHvbWa aWbaaSqabeaacqGHsislcaWGubaaaOGaaCyvamaaCaaaleqabaGaaG OmaaaakiaahwfadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqp [email protected]@ .

·         Given that U and R exist we can write RU=[ RU R T ]R [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHsbGaeyyXIC TaaCyvaiabg2da9maadmaabaGaaCOuaiabgwSixlaahwfacqGHflY1 caWHsbWaaWbaaSqabeaacaWGubaaaaGccaGLBbGaayzxaaGaeyyXIC [email protected]@  so if we define V=RU R T [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHwbGaeyypa0 [email protected]@  then A=VR [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbGaeyypa0 [email protected]@ .   It is easy to show that V is symmetric.

·         To see that the decomposition is unique, suppose that A= R ^ U ^ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbGaeyypa0 [email protected]@  for some other tensors R ^ , U ^ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWHsbGbaKaaca [email protected]@ .  Then A T A= U ^ 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbWaaWbaaS qabeaacaWGubaaaOGaaCyqaiabg2da9iqahwfagaqcamaaCaaaleqa [email protected]@ .  But A T A [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbWaaWbaaS [email protected]@  has a unique square root so U ^ =U [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWHvbGbaKaacq [email protected]@ .  The uniqueness of R follows immediately.