Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 356

In the section before last, Sect. A.5 on Linear Transformations, the eigenvalue
problem for a linear transformation r t was considered. Here we extend those

results by considering r and t to be vectors and A to be a symmetric second order
tensor, A

¼ A

A T . The problem is actually little changed until its conclusion. The
eigenvalues are still given by (A.52) or, for n

¼

¼

3 by (A.54). The values of three
quantities I A , II A , III A , defined by (A.55), (A.56), (A.57) are the same except that
A 12 ¼ A T .

These quantities may now be called the invariants of the tensor A since their value
is the same independent of the coordinate system chosen for their determination. As
an example of the invariance with respect to basis, this property will be derived for
I A ¼

A 21 , A 13 ¼ A 31 and A 32 ¼ A 23 due to the assumed symmetry of A , A ¼

tr A . Let T ¼ A in (A.86), then set the indices k ¼ m and sum from one to n

over the index k , thus

A kk ¼ T ab Q k a Q k b ¼ A ab d ab ¼ A aa (A.88)

The transition across the second equal sign is a simple rearrangement of terms.
The transition across the second equal sign is based on the condition

Q k a Q k b ¼ d ab (A.89)

which is an alternate form of (A.67), a form equivalent to Q T 1 . The transition

across the fourth equal sign employs the definition of the Kronecker delta and the
summation over

Q ¼

b

. The result is that the trace of the matrix of second order tensor
components relative to any basis is the same number,

A kk ¼ A aa : (A.90)

It may also be shown that II A and III A are invariants of the tensor A .
Example A.7.2 (An Extension of Example A.5.5)

Consider the matrix given by (A.58) in Example A.5.5 to be the components of a
tensor. Construct the eigenvectors of that tensor and use those eigenvectors to
construct an eigenbasis

2 3

18 6 6
6 50
601

4 5 :

A ¼ ð A : 58 Þ repeated

Solution: The eigenvalues were shown to be 27, 18, and 9. It can be shown that the
eigenvalues must always be real numbers if A is symmetric. Eigen n-tuples were
constructed using these eigenvalues. The first eigen n-tuple was obtained by
substitution of (A.58) and

l ¼

27 into (A.49), thus

9 t 1 þ 6 t 2 þ 6 t 3 ¼ 0 ; 6 t 1 12 t 2 ¼ 0 ; 6 t 1 6 t 3 ¼ 0 : (A.60) repeated