Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 397

Orthogonal Transformations

The matrix Q [ Q i a ] characterizing the change from the Latin orthonormal basis

e i in an N-dimensional vector space to the Greek basis e a (or vice versa) is a special
type of linear transformation called an orthogonal transformation. Taking the scalar
product of e i with e j where e i and e j both have the representation (A.62),

¼

e i ¼ Q i a e a and e j ¼ Q j b e b : (A.66)

it follows that

e i e j ¼ d ij ¼ Q i a Q j b e a e b ¼ Q i a Q j b d ab ¼ Q i a Q j a : (A.67)

There are a number of steps in the calculation (A.67) that should be considered
carefully. First, the condition of orthonormality of the bases has been used twice, e i

e b ¼ d ab . Second, the transition from the term before the last equal
sign to the term after that sign is characterized by a change from a double sum to a
single sum over n and the loss of the Kronecker delta

e j ¼ d ij and e a

d ab . This occurs because the sum

over b in the double sum is always zero except in the special case when a ¼ b due to

the presence of the Kronecker delta d ab . Third, a comparison of the last term in (A.67)

with the definition of matrix product (A.20) suggests that it is a matrix product of Q
with itself. However, a careful comparison of the last term in (A.67) with the definition
of matrix product (A.20) shows that the summation is over a different index in the
second element of the product. In order for the last term in (A.67) to represent a matrix
product, the a index should appear as the first subscripted index rather than the second.

However, this index may be relocated in the secondmatrix by using the transposition

operation. Thus the last term in equation (A.67) is the matrix product of Q with Q T as
may be seen from the first of equations (A.18). Thus, since the matrix of Kronecker
delta components is the unit matrix 1 , it has been shown that

a

Q T 1 ¼ Q : (A.68)

A transformation Q satisfying (A.68) is to be an orthogonal transformation , its
inverse is equal to its transpose, Q 1 Q T . ¼

Proof of Invariance for the Trace of a Matrix

As an example of the invariance with respect to basis, this property will be derived
for I 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)