Page 11-4
Vector-matrix multiplication, on the other hand, is not defined. This
multiplication can be performed, however, as a special case of matrix
multiplication as defined next.
Matrix multiplication
Matrix multiplication is defined by
C
m
×
n
=
A
m
×
p
⋅
B
p
×
n
, where
A
= [a
ij
]
m
×
p
,
B
=
[b
ij
]
p
×
n
, and
C
= [c
ij
]
m
×
n
. Notice that matrix multiplication is only possible if the
number of columns in the first operand is equal to the number of rows of the
second operand. The general term in the product, c
ij
, is defined as
.
,
,
2
,
1
;
,
,
2
,
1
,
1
n
j
m
i
for
b
a
c
p
k
kj
ik
ij
K
K
=
=
⋅
=
∑
=
This is the same as saying that the element in the i-th row and j-th column of
the product,
C
, results from multiplying term-by-term the i-th row of
A
with the j-
th column of
B
, and adding the products together. Matrix multiplication is not
commutative, i.e., in general,
A
⋅
B
≠
B
⋅
A
. Furthermore, one of the
multiplications may not even exist. The following screen shots show the results
of multiplications of the matrices that we stored earlier:
The matrix-vector multiplication introduced in the previous section can be
thought of as the product of a matrix m
×
n with a matrix n
×
1 (i.e., a column
vector) resulting in an m
×
1 matrix (i.e., another vector). To verify this
assertion check the examples presented in the previous section. Thus, the
vectors defined in Chapter 9 are basically column vectors for the purpose of
matrix multiplication.
