Invariants of tensors
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the fields of
multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...
and
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, the principal invariants of the second rank
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
\mathbf are the coefficients of the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
:\ p(\lambda)=\det (\mathbf-\lambda \mathbf), where \mathbf is the identity operator and \lambda_i \in\mathbb represent the polynomial's
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
. More broadly, any scalar-valued function f(\mathbf) is an invariant of \mathbf if and only if f(\mathbf\mathbf\mathbf^T)=f(\mathbf) for all orthogonal \mathbf. This means that a formula expressing an invariant in terms of components, A_, will give the same result for all Cartesian bases. For example, even though individual diagonal components of \mathbf will change with a change in basis, the sum of diagonal components will not change.


Properties

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

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

For such tensors, the principal invariants are given by: : \begin I_1 &= \mathrm(\mathbf) = A_+A_+A_ = \lambda_1+\lambda_2+\lambda_3 \\ I_2 &= \frac \left( \mathrm \left( \mathbf^2 \right) - (\mathrm(\mathbf))^2 \right) = A_A_+A_A_+A_A_-A_A_-A_A_-A_A_ = \lambda_1 \lambda_2 + \lambda_1 \lambda_3 + \lambda_2 \lambda_3 \\ I_3 &= \det (\mathbf)= -A_ A_ A_ + A_ A_ A_ + A_ A_ A_ - A_ A_ A_ - A_ A_ A_ + A_ A_ A_ = \lambda_1 \lambda_2 \lambda_3 \end For symmetric tensors, these definitions are reduced. The correspondence between the principal invariants and the characteristic polynomial of a tensor, in tandem with the
Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies it ...
reveals that :\ \mathbf^3 - I_1 \mathbf^2 +I_2 \mathbf -I_3 \mathbf= 0 where \mathbf is the second-order identity tensor.


Main invariants

In addition to the principal invariants listed above, it is also possible to introduce the notion of main invariants : \begin J_1 &= \lambda_1 + \lambda_2 + \lambda_3 = I_1 \\ J_2 &= \lambda_1^2 + \lambda_2^2 + \lambda_3^2 = I_1^2 - 2 I_2 \\ J_3 &= \lambda_1^3 + \lambda_2^3 + \lambda_3^3 = I_1^3 - 3 I_1 I_2 + 3 I_3 \end which are functions of the principal invariants above. These are the coefficients of the characteristic polynomial of the deviator \mathbf - (\mathrm(\mathbf)/3)\mathbf, 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

Furthermore, mixed invariants between pairs of rank two tensors may also be defined.


Calculation of the invariants of order two tensors of higher dimension

These may be extracted by evaluating the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
directly, using the Faddeev-LeVerrier algorithm for example.


Calculation of the invariants of higher order tensors

The invariants of rank three, four, and higher order tensors may also be determined.


Engineering applications

A scalar function f that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density, or
Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal In thermodynamics, an isotherma ...
, of a nonlinear material possessing isotropic symmetry. This technique was first introduced into isotropic
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
by Howard P. Robertson in 1940 where he was able to derive
Kármán–Howarth equation In isotropic turbulence the Kármán–Howarth equation (after Theodore von Kármán and Leslie Howarth 1938), which is derived from the Navier–Stokes equations, is used to describe the evolution of non-dimensional longitudinal autocorrelation. ...
from the invariant principle.
George Batchelor George Keith Batchelor FRS (8 March 1920 – 30 March 2000) was an Australian applied mathematician and fluid dynamicist. He was for many years a Professor of Applied Mathematics in the University of Cambridge, and was founding head of the De ...
and
Subrahmanyan Chandrasekhar Subrahmanyan Chandrasekhar (; ) (19 October 1910 – 21 August 1995) was an Indian-American theoretical physicist who spent his professional life in the United States. He shared the 1983 Nobel Prize for Physics with William A. Fowler for "... ...
exploited this technique and developed an extended treatment for axisymmetric turbulence.


Invariants of non-symmetric tensors

A real tensor \mathbf in 3D (i.e., one with a 3x3 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 \mathbf are : = \begin 931 & 5480 & -717\\ -5120 & 1650 & 1090\\ 1533 & -610 & 1169 \end, the first step would be to evaluate the axial vector \mathbf associated with the skew-symmetric part. Specifically, the axial vector has components :\begin w_1&=\frac=-850\\ w_2&=\frac=-1125\\ w_3&=\frac=-5300 \end The next step finds the principal values of the symmetric part of \mathbf. Even though the eigenvalues of a real non-symmetric tensor might be complex, the eigenvalues of its 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 \mathbf points within the first octant. With respect to that special basis, the components of \mathbf are : '= \begin 1875 & -2500 & 3125\\ 2500 & 1250 & -3750\\ -3125 & 3750 & 625 \end, The first three invariants of \mathbf are the diagonal components of this matrix: a_1=A'_=1875 , a_2=A'_=1250, a_3=A'_=625 (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'_1=A'_=3750, w'_2=A'_=3125, w'_3=A'_=2500. Note: the magnitude of the axial vector, \sqrt, is the sole invariant of the skew part of \mathbf, whereas these distinct three invariants characterize (in a sense) "alignment" between the symmetric and skew parts of \mathbf. Incidentally, it is a myth that a tensor is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
if its eigenvalues are positive. Instead, it is positive definite if and only if the eigenvalues of its symmetric part are positive.


See also

*
Symmetric polynomial In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...
*
Elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sym ...
*
Newton's identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomia ...
*
Invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...


References

{{reflist, 30em Tensors Invariant theory Linear algebra