Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 349

sets, e i and e a , taken together are linearly dependent and therefore we can write that
e i is a linear combination of the e a 's and vice versa. These relationships are
expressed as linear transformations,

Q 1
i

e i ¼ Q i a e a and e a ¼ e i ; (A.62)

a

where Q [ Q i a ] is the matrix characterizing the linear transformation. For unam-

biguous conversion between the index and matrix notation the Latin index is fixed
in the place of the row index (i.e., first) and the Greek index is frozen in the place of
the column index (i.e., second) in the notation employed here. In the case of n

¼

¼ 3

the first of these equations may be expanded into a system of three equations:

e 1 ¼ Q 1I e I þ Q 1II e II þ Q 1III e III ;

e 2 ¼ Q 2I e I þ Q 2II e II þ Q 2III e III ;

e 3 ¼ Q 3I e I þ Q 3II e II þ Q 3III e III : (A.63)

If one takes the scalar product of e I with each of these equations and notes that
since the e a ,

a ¼

I, II, III, form an orthonormal basis, then e I

e II ¼ e I e III ¼

0, and

Q 1I ¼ e 3 . Repeating the

scalar product operation for e II and e III shows that, in general, Q i a ¼

e 1 e I ¼ e I e 1 , Q 2I ¼ e 2 e I ¼ e I e 2 , and Q 3I ¼ e 3 e I ¼ e I

e i .
Recalling that the scalar product of two vectors is the product of magnitudes of each
vector and the cosine of the angle between the two vectors (A.61), and that the base
vectors are unit vectors, it follows that Q i a ¼

e i e a ¼ e a

e i are just the cosines of
angles between the base vectors of the two bases involved. Thus the components of
the linear transformation Q

e i e a ¼ e a

[ Q i a ] are the cosines of the angles between the base
vectors of the two bases involved. Because the definition of the scalar product
(A.61) is valid in n dimensions, all these results are valid in n dimensions even
though the two- and three-dimensional geometric interpretation of the components
of the linear transformation Q as the cosines of the angles between coordinate axes
is no longer valid.
The geometric analogy is very helpful, so considerations in three dimensions are
continued. Three-dimensional Greek and Latin coordinate systems are illustrated
on the left-hand side of Fig. A.1 . The matrix Q with components Q i a ¼

¼

e a relates
the components of vectors and base vectors associated with the Greek system to
those associated with the Latin system,

e i

2 3

e 1 e I e 1 e II e 1 e III

4 5 :

Q ¼½ Q i a ¼½ e i e a ¼ e 2 e I e 2 e II e 2 e III (A.64)

e 3 e I e 3 e II e 3 e III

In the special case when the e 1 and e I are coincident, the relative rotation
between the two observers' frames is a rotation about that particular selected and
fixed axis, and the matrix Q has the special form