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**Eigenvalues, eigenvectors of tensors**

The scalar Î»i characterize eigenvalues (or principal values) of A if there exist corresponding nonzero normalized eigenvectors (or principal directions or principal axes) of A, so that

To identify the eigenvectors of a tensor, we use subsequently a hat on the vector quantity concerned, for example, .

Thus, a set of homogeneous algebraic equations for the unknown eigenvalues Î»_{i} , i = 1,2,3, and the unknown eigenvectors , i = 1,2,3, is

(A âˆ’ Î»_{i}1) = o, (i = 1,2,3; no summation). (2.137)

For the above system to have a solution â‰ **o** the determinant of the system must vanish. Thus,

det(A âˆ’ Î»_{i}1) = 0 (2.138)

where

This requires that we solve a cubic equation in Î», usually written as

called the characteristic polynomial (or equation) for **A,** the solutions of which are the eigenvalues Î»_{i} , i = 1,2,3. Here, I_{i} , i = 1,2,3, are the so-called principal scalar invariants of A and are given by

If **A** is invertible then we can compute I_{2} using the expression I_{2} = tr(A^{âˆ’1} ) det(**A**). A repeated application of tensor A to equation (2.136) yields i = 1,2,3, for any positive integer Î±. (If A is invertible then Î± can be any integer; not necessarily positive.) Using this relation and (2.140) multiplied by , we obtain the well-known Cayley-Hamilton equation:

It states that every second order tensor **A** satisfies its own characteristic equation. As a consequence of Cayley-Hamilton equation, we can express **A ^{Î±}** in terms of

For a symmetric tensor

Any symmetric tensor S may be represented by its eigenvalues Î»

(2.143)

called the spectral representation (or spectral decomposition) of S. Thus, when orthonormal eigenvectors are used as the Cartesian basis to represent S then

(2.144)

The above holds when all the three eigenvalues are distinct. On the other hand, if there exists a pair of equal roots, i.e., Î»

where and denote projection tensors introduced in (2.109) and (2.110) respectively. Finally, if all the three eigenvalues are equal, i.e., Î»

S = Î»1, (2.146)

where every direction is a principal direction and every set of mutually orthogonal basis denotes principal axes.

It is important to recognize that eigenvalues characterize the physical nature of the tensor and that they do not depend on the coordinates chosen.

Square root theorem Let C be symmetric and positive definite tensor. Then there is a unique positive definite, symmetric tensor U such that

U

We write for U. If spectral representation of C is:

(2.148)

then the spectral representation for U is:

(2.149)

where we have assumed that the eigenvalues of C, Î»

(2.150)

If Î»

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