# Invariants of tensors – Wikipedia

In mathematics, in the fields of multilinear algebra and representation theory, the **principal invariants** of the second rank tensor

^{[1]}

- p(λ)=det(A−λI){displaystyle p(lambda )=det(mathbf {A} -lambda mathbf {I} )} ,

where

I{displaystyle mathbf {I} }λi∈C{displaystyle lambda _{i}in mathbb {C} } is the identity operator and

represent the polynomial’s eigenvalues.

More broadly, any scalar-valued function

f(A){displaystyle f(mathbf {A} )}is an invariant of

f(QAQT)=f(A){displaystyle f(mathbf {Q} mathbf {A} mathbf {Q} ^{T})=f(mathbf {A} )} if and only if

Q{displaystyle mathbf {Q} } for all orthogonal

Aij{displaystyle A_{ij}} . This means that a formula expressing an invariant in terms of components,

A{displaystyle mathbf {A} } , will give the same result for all Cartesian bases. For example, even though individual diagonal components of

will change with a change in basis, the sum of diagonal components will not change.

## Properties[edit]

The principal invariants do not change with rotations of the coordinate system (they are objective, or in more modern terminology, satisfy the principle of material frame-indifference) and any function of the principal invariants is also objective.

## Calculation of the invariants of rank two tensors[edit]

In a majority of engineering applications, the principal invariants of (rank two) tensors of dimension three are sought, such as those for the right Cauchy-Green deformation tensor.

### Principal invariants[edit]

For such tensors, the principal invariants are given by:

- I1=tr(A)=A11+A22+A33=λ1+λ2+λ3I2=12((tr(A))2−tr(A2))=A11A22+A22A33+A11A33−A12A21−A23A32−A13A31=λ1λ2+λ1λ3+λ2λ3I3=det(A)=−A13A22A31+A12A23A31+A13A21A32−A11A23A32−A12A21A33+A11A22A33=λ1λ2λ3{displaystyle {begin{aligned}I_{1}&=mathrm {tr} (mathbf {A} )=A_{11}+A_{22}+A_{33}=lambda _{1}+lambda _{2}+lambda _{3}\I_{2}&={frac {1}{2}}left((mathrm {tr} (mathbf {A} ))^{2}-mathrm {tr} left(mathbf {A} ^{2}right)right)=A_{11}A_{22}+A_{22}A_{33}+A_{11}A_{33}-A_{12}A_{21}-A_{23}A_{32}-A_{13}A_{31}=lambda _{1}lambda _{2}+lambda _{1}lambda _{3}+lambda _{2}lambda _{3}\I_{3}&=det(mathbf {A} )=-A_{13}A_{22}A_{31}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}-A_{11}A_{23}A_{32}-A_{12}A_{21}A_{33}+A_{11}A_{22}A_{33}=lambda _{1}lambda _{2}lambda _{3}end{aligned}}}

For symmetric tensors, these definitions are reduced.^{[2]}

The correspondence between the principal invariants and the characteristic polynomial of a tensor, in tandem with the Cayley–Hamilton theorem reveals that

- A3−I1A2+I2A−I3I=0{displaystyle mathbf {A} ^{3}-I_{1}mathbf {A} ^{2}+I_{2}mathbf {A} -I_{3}mathbf {I} =0}

where

I{displaystyle mathbf {I} }is the second-order identity tensor.

### Main invariants[edit]

In addition to the principal invariants listed above, it is also possible to introduce the notion of main invariants^{[3]}^{[4]}

- J1=λ1+λ2+λ3=I1J2=λ12+λ22+λ32=I12−2I2J3=λ13+λ23+λ33=I13−3I1I2+3I3{displaystyle {begin{aligned}J_{1}&=lambda _{1}+lambda _{2}+lambda _{3}=I_{1}\J_{2}&=lambda _{1}^{2}+lambda _{2}^{2}+lambda _{3}^{2}=I_{1}^{2}-2I_{2}\J_{3}&=lambda _{1}^{3}+lambda _{2}^{3}+lambda _{3}^{3}=I_{1}^{3}-3I_{1}I_{2}+3I_{3}end{aligned}}}

which are functions of the principal invariants above. These are the coefficients of the characteristic polynomial of the deviator

A−(tr(A)/3)I{displaystyle mathbf {A} -(mathrm {tr} (mathbf {A} )/3)mathbf {I} }, such that it is traceless. The separation of a tensor into a component that is a multiple of the identity and a traceless component is standard in hydrodynamics, where the former is called isotropic, providing the modified pressure, and the latter is called deviatoric, providing shear effects.

### Mixed invariants[edit]

Furthermore, mixed invariants between pairs of rank two tensors may also be defined.^{[4]}

## Calculation of the invariants of order two tensors of higher dimension[edit]

These may be extracted by evaluating the characteristic polynomial directly, using the Faddeev-LeVerrier algorithm for example.

## Calculation of the invariants of higher order tensors[edit]

The invariants of rank three, four, and higher order tensors may also be determined.^{[5]}

## Engineering applications[edit]

A scalar function

f{displaystyle f}^{[6]}

This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940 where he was able to derive Kármán–Howarth equation from the invariant principle.^{[7]}George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence.^{[8]}^{[9]}^{[10]}

### Invariants of non-symmetric tensors[edit]

A real tensor

A{displaystyle mathbf {A} }A{displaystyle mathbf {A} } in 3D (i.e., one with a 3×3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part. For example, if the Cartesian components of

are

- [A]=[9315480−717−5120165010901533−6101169],{displaystyle [A]={begin{bmatrix}931&5480&-717\-5120&1650&1090\1533&-610&1169end{bmatrix}},}

the first step would be to evaluate the axial vector

w{displaystyle mathbf {w} }associated with the skew-symmetric part. Specifically, the axial vector has components

- w1=A32−A232=−850w2=A13−A312=−1125w3=A21−A122=−5300{displaystyle {begin{aligned}w_{1}&={frac {A_{32}-A_{23}}{2}}=-850\w_{2}&={frac {A_{13}-A_{31}}{2}}=-1125\w_{3}&={frac {A_{21}-A_{12}}{2}}=-5300end{aligned}}}

The next step finds the principal values of the symmetric part of

A{displaystyle mathbf {A} }*symmetric part* will always be real and therefore can be ordered from largest to smallest. The corresponding orthonormal principal basis directions can be assigned senses to ensure that the axial vector

A{displaystyle mathbf {A} } points within the first octant. With respect to that special basis, the components of

are

- [A′]=[1875−2500312525001250−3750−31253750625],{displaystyle [A’]={begin{bmatrix}1875&-2500&3125\2500&1250&-3750\-3125&3750&625end{bmatrix}},}

The first three invariants of

A{displaystyle mathbf {A} }a1=A11′=1875,a2=A22′=1250,a3=A33′=625{displaystyle a_{1}=A’_{11}=1875,a_{2}=A’_{22}=1250,a_{3}=A’_{33}=625} are the diagonal components of this matrix:

w1′=A32′=3750,w2′=A13′=3125,w3′=A21′=2500{displaystyle w’_{1}=A’_{32}=3750,w’_{2}=A’_{13}=3125,w’_{3}=A’_{21}=2500} (equal to the ordered principal values of the tensor’s symmetric part). The remaining three invariants are the axial vector’s components in this basis:

w⋅w{displaystyle {sqrt {mathbf {w} cdot mathbf {w} }}} . Note: the magnitude of the axial vector,

A{displaystyle mathbf {A} } , is the sole invariant of the skew part of

A{displaystyle mathbf {A} } , whereas these distinct three invariants characterize (in a sense) “alignment” between the symmetric and skew parts of

*symmetric part* are positive.

## See also[edit]

## References[edit]

**^**Spencer, A. J. M. (1980).*Continuum Mechanics*. Longman. ISBN 0-582-44282-6.**^**Kelly, PA. “Lecture Notes: An introduction to Solid Mechanics” (PDF). Retrieved 27 May 2018.**^**Kindlmann, G. “Tensor Invariants and their Gradients” (PDF). Retrieved 24 Jan 2019.- ^
^{a}^{b}Schröder, Jörg; Neff, Patrizio (2010).*Poly-, Quasi- and Rank-One Convexity in Applied Mechanics*. Springer. **^**Betten, J. (1987). “Irreducible Invariants of Fourth-Order Tensors”.*Mathematical Modelling*.**8**: 29–33. doi:10.1016/0270-0255(87)90535-5.**^**Ogden, R. W. (1984).*Non-Linear Elastic Deformations*. Dover.**^**Robertson, H. P. (1940). “The Invariant Theory of Isotropic Turbulence”.*Mathematical Proceedings of the Cambridge Philosophical Society*. Cambridge University Press.**36**(2): 209–223. Bibcode:1940PCPS…36..209R. doi:10.1017/S0305004100017199.**^**Batchelor, G. K. (1946). “The Theory of Axisymmetric Turbulence”.*Proc. R. Soc. Lond. A*.**186**(1007): 480–502. Bibcode:1946RSPSA.186..480B. doi:10.1098/rspa.1946.0060.**^**Chandrasekhar, S. (1950). “The Theory of Axisymmetric Turbulence”.*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**242**(855): 557–577. Bibcode:1950RSPTA.242..557C. doi:10.1098/rsta.1950.0010. S2CID 123358727.**^**Chandrasekhar, S. (1950). “The Decay of Axisymmetric Turbulence”.*Proc. R. Soc. A*.**203**(1074): 358–364. Bibcode:1950RSPSA.203..358C. doi:10.1098/rspa.1950.0143. S2CID 121178989.

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