Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 342

property that A applied to the sum ( r þ t ) follows a distributive law A ð r þ t Þ

¼ A r þ A t and that multiplication by a scalar a follows the rule að A r Þ¼

A ða r Þ . These two properties may be combined into one, A ða r þ b t Þ¼a A r

þb are scalars. The composition of linear transformations is

again a linear transformation. Consider the linear transformation t

A t where a and b

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 u . The result of the composition of the two linear transformations,

r ¼ At and t ¼ Bu , is then a new linear transformation r ¼ Cu where the square
matrix C is given by the matrix product AB . 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 C

¼ B u in the indicial notation,

t k ¼ B km u m ; (A.42)

is substituted into r ¼ A t 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. The calculation will be similar to the one above with the
exception that the summation symbols will appear.

¼ A

Example A.5.1

Determine the result r ¼ Cu of the composition of the two linear transformations,
r ¼ A t and t ¼ B u , where A and B are given by

2
4

3
5 ;

2
4

3
5 :

123
456
789

10 11 12

A ¼ B ¼ 13 14 15

16 17 18

Solution: The matrix product AB yields the square matrix C representing the
composed linear transformation,

2
4

3
5 :

84 90 96

A B ¼ 201 216 231

318 342 366