Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

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Civil Engineering (CE) : Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

The document Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Advanced Solid Mechanics - Notes, Videos, MCQs & PPTs.
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Eigenvalues, eigenvectors of tensors
The scalar λi characterize eigenvalues (or principal values) of A if there exist corresponding nonzero normalized eigenvectors Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev (or principal directions or principal axes) of A, so that

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

To identify the eigenvectors of a tensor, we use subsequently a hat on the vector quantity concerned, for example, Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev .
Thus, a set of homogeneous algebraic equations for the unknown eigenvalues λi , i = 1,2,3, and the unknown eigenvectors Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev , i = 1,2,3, is
(A − λi1)Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev = o, (i = 1,2,3; no summation).                                            (2.137)

For the above system to have a solution Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev ≠ o the determinant of the system must vanish. Thus,

det(A − λi1) = 0                                                                                        (2.138)

where

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

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

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

called the characteristic polynomial (or equation) for A, the solutions of which are the eigenvalues λi , i = 1,2,3. Here, Ii , i = 1,2,3, are the so-called principal scalar invariants of A and are given by

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

If A is invertible then we can compute I2 using the expression I2 = tr(A−1 ) det(A). A repeated application of tensor A to equation (2.136) yields Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRevEigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev i = 1,2,3, for any positive integer α. (If A is invertible then α can be any integer; not necessarily positive.) Using this relation and (2.140) multiplied by Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev, we obtain the well-known Cayley-Hamilton equation:

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

It states that every second order tensor A satisfies its own characteristic equation. As a consequence of Cayley-Hamilton equation, we can express Aα in terms of A2 , A, and principal invariants for positive integer, α > 2. (If A is invertible, the above holds for any integer value of α positive or negative provided α ≠ {0, 1, 2}.)

For a symmetric tensor S the characteristic equation (2.140) always has three real solutions and the set of eigenvectors form a orthonormal basis {nˆi} (the proof of this statement is omitted). Hence, for a positive definite symmetric tensor A, all eigenvalues λi are (real and) positive since, using (2.136), we have λi =Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev � Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev > 0, i = 1,2,3.

Any symmetric tensor S may be represented by its eigenvalues λ, i = 1,2,3, and the corresponding eigenvectors of S forming an orthonormal basis {Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev}. Thus, S can be expressed as

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev                           (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
Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev                        (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., λ1 = λ2 ≠  λ3, with an unique eigenvector Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev associated with λ3, we deduce that

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

where Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev  and Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev denote projection tensors introduced in (2.109) and (2.110) respectively. Finally, if all the three eigenvalues are equal, i.e., λ1 = λ= λ3 = λ, then

S = λ1,                                                                              (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

U2 = C.                            (2.147)

We write Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev for U. If spectral representation of C is:

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev                         (2.148)


then the spectral representation for U is:

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev                                           (2.149)

where we have assumed that the eigenvalues of C, λare distinct. On the other hand if λ1 = λ2 = λ3 = λ then

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev (2.150)

If λ1 = λ2 = λ ≠ λ3, with an unique eigenvector Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev associated with λ3 then

Eigenvalues, Eigenvectors of Tensors Civil Engineering (CE) Notes | EduRev

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