Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 351

equations (A.18). Thus, since the matrix of Kronecker delta components is the unit
matrix 1 , it has been shown that

Q T 1 ¼ Q : (A.68)

If we repeat the calculation of the scalar product, this time using e a and e b rather
than e i and e j, then it is found that 1 Q T ¼ Q and, combined with the previous result,

Q T Q T 1 ¼ Q ¼ Q : (A.69)

Using the fact that Det 1 ¼ 1, and two results that are proved in Sect. A.8,

Det A T , it follows from 1 Q T or 1 Det A B ¼ Det A Det B , and Det A ¼ ¼ Q ¼

Q T Q that Q is non-singular and Det Q ¼ 1. Comparing the matrix equations

Q T and 1 Q T Q 1 1 ¼ Q ¼ Q with the equations defining the inverse of Q , 1 ¼ Q

Q 1

¼ Q , it follows that

Q 1 Q T ¼ ; (A.70)

since the inverse exists (Det Q is not singular) and is unique. Any matrix Q that
satisfies equation (A.69) is called an orthogonal matrix. Any change of orthonormal
bases is characterized by an orthogonal matrix and is called an orthogonal transfor-
mation. Finally, since Q 1 Q T the representations of the transformation of bases ¼

(A.62) may be rewritten as

e i ¼ Q i a e a and e a ¼ Q i a e i :

(A.71)

Orthogonal matrices are very interesting, useful and easy to handle; their deter-
minant is always plus or minus one and their inverse is obtained simply by comput-
ing their transpose. Furthermore, the multiplication of orthogonal matrices has the
closure property. To see that the product of two n by n orthogonal matrices is another
n by n orthogonal matrix, let R and Q be orthogonal matrices and consider their
product denoted by W

Q . The inverse of W is given by W 1 Q 1 R 1 and

¼ R ¼

its transpose by W T Q T R T . Since R and Q are orthogonal matrices, Q 1 R 1

¼ ¼

Q T W T and therefore W is orthogonal. It follows then

that the set of all orthogonal matrices has the closure property as well as the
associative property with respect to the multiplication operation, an identity element
(the unit matrix 1 is orthogonal), and an inverse for each member of the set.
Here we shall consider changing the basis to which a given vector is referred.
While the vector v itself is invariant with respect to a change of basis, the
components of v will change when the basis to which they are referred is changed.
The components of a vector v referred to a Latin basis are then v i , i

R T , it follows that W 1

¼

¼ 1, 2, 3, ... , n ,

while the components of the same vector referred to a Greek basis are v a , a ¼ I, II,

III, ... , n . Since the vector v is unique,

v ¼ v i e i ¼ v a e a : (A.72)