Page 112 - Matrix Analysis & Applied Linear Algebra

106 Chapter 3 Matrix Algebra

Use this along with the left-hand distributive property to write
q q

B ∗k c kj =
[A(BC)] ij = A i∗ [BC] ∗j = A i∗ A i∗ B ∗k c kj
k=1 k=1
q

= [AB] ik c kj =[AB] i∗ C ∗j =[(AB)C] ij .
k=1
Example 3.6.1
Linearity of Matrix Multiplication. Let A be an m × n matrix, and f be
the function defined by matrix multiplication
f(X n×p )= AX.
The left-hand distributive property guarantees that f is a linear function be-
cause for all scalars α and for all n × p matrices X and Y,
f(αX + Y)= A(αX + Y)= A(αX)+ AY = αAX + AY
= αf(X)+ f(Y).
Of course, the linearity of matrix multiplication is no surprise because it was
the consideration of linear functions that motivated the definition of the matrix
product at the outset.

For scalars, the number 1 is the identity element for multiplication because
it has the property that it reproduces whatever it is multiplied by. For matrices,
there is an identity element with similar properties.

Identity Matrix
The n × n matrix with 1’s on the main diagonal and 0’s elsewhere

10 ··· 0
 
 01 ··· 0 
I n =  . . . . 
 . . .
. . . . . 
00 ··· 1
is called the identity matrix of order n. For every m × n matrix A,

AI n = A and I m A = A.

The subscript on I n is neglected whenever the size is obvious from the
context.