Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 359

l 1 6¼ l 2 , then n and m are perpendicular. If the two eigenvalues are not
distinct, then any vector in a plane is an eigenvector so that one can always
construct a mutually orthogonal set of eigenvectors for a symmetric matrix.
Generalizing Example A.7.2 above from 3 to n it may be concluded that any n by
n matrix of symmetric tensor components A has a representation in which the
eigenvalues lie along the diagonal of the matrix and the off-diagonal elements
are all zero. The last expression in (A.96) is a particular example of this when
n

Thus, if

¼ 3. If the symmetric tensor A has n eigenvalues l i , then a quadratic form c may

be formed from A and a vector n-tuple x , thus

2

c ¼ x A x ¼ l i ð x i Þ : (A.102)

If all the eigenvalues of A are positive, this quadratic form is said to be positive
definite and

2

c ¼ x A x ¼ l i ð x i Þ > 0 for all x 6¼ 0 : (A.103)

(If all the eigenvalues of A are negative the quadratic form is said to be negative
definite .) Transforming the tensor A to an arbitrary coordinate system the equation
(A.102) takes the form

c ¼ x A x ¼ A ij x i x i > 0 for all x 6¼ 0 : (A.104)

A tensor A with the property (A.104), when it is used as the coefficients of a
quadratic form, is said to be positive definite. In the mechanics of materials there
are a number of tensors that are positive definite due to physics they represent. The
moment of inertia tensor is an example. Others will be encountered as material
coefficients in constitutive equations in Chap. 5 .

Problems

A.7.1 Consider two three-dimensional coordinate systems. One coordinate system
is a right-handed coordinate system. The second coordinate system is
obtained from the first by reversing the direction of the first ordered base
vector and leaving the other two base vectors to be identical with those in the
first coordinate system. Show that the orthogonal transformation relating
these systems is given by

2 3

100
010
001

4 5

Q ¼

1.
A.7.2 Construct the eigenvalues and the eigenvectors of the matrix T of tensor
components where

and that its determinant is