Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 396

Introduction of the free index convention condenses this system of equations
vertically,

r i ¼ A ik t k : (A.38)

This result may also be represented in the matrix notation as a combination of n-
tuples, r and t , and a square matrix A ,

r ¼ A t; (A.39)

where the dot between A and t indicates that the summation is with respect to one
index of A and one index of t .

The Composition of Linear Transformations

The composition of linear transformations is again a linear transformation. Consider
the linear transformation t t (meaning u is transformed into t ) which is

combined with the linear transformation (A.39) r

¼ B u , u !

¼ A t , t ! r to transform u ! r ,

thus r ¼ A B u , and if we let C A B , then r ¼ C u . The result of the composition

of the two linear transformations, r ¼ A t and t ¼ B u , is then a new linear

transformation r ¼ C u where the square matrix C is given by the matrix product

A B . To verify that it is, in fact, a matrix multiplication, the composition of

transformations is done again in the indicial notation. The transformation t ¼ B u

in the indicial notation,

t k ¼ B km u m ;

(A.42)

is substituted into r ¼ At in the indicial notation (A.38),

r i ¼ A ik B km u m ; (A.43)

which may be rewritten as

r i ¼ C im u m ; (A.44)

where C is defined by:

C im ¼ A ik B km : (A.45)

Comparison of (A.45) with (A.20) shows that C is the matrix product of A and B ,

C B . The calculation from (A.42) to (A.45) may be repeated using the Einstein

summation convention.

¼ A