Biomedical Engineering Reference
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B
11
B
12
B
21
B
22
A
11
A
12
A
21
A
22
A
B
¼
A
11
B
11
þ
A
12
B
21
A
11
B
12
þ
A
12
B
22
¼
;
(A.22)
A
21
B
11
þ
A
22
B
21
A
21
B
12
þ
A
22
B
22
where, in this case, the products (A.20) and (A.21) stand for the
n
2
2
2
4
separate equations, the right-hand sides of which are the four elements of the last
matrix in (A.22).
The dot between the matrix product
A
¼
¼
B
indicates that one index from
A
and one
index from
B
is to be summed over. The positioning of the summation index on the
two matrices involved in a matrix product is critical and is reflected in the matrix
notation by the transpose. In the three equations below, (A.21), study carefully how
the positions of the summation indices within the summation sign change in relation
to the position of the transpose on the matrices in the associated matrix product:
B
T
A
T
A
T
B
T
ð
A
Þ
ij
¼
A
ik
B
jk
; ð
B
Þ
ij
¼
A
ki
B
kj
; ð
Þ
ij
¼
A
ki
B
jk
:
(A.23)
A very significant feature of matrix multiplication is noncommutatively, that is
to say
A
B
6¼
B
A
. Note, for example, the transposed product
B
A
of the multipli-
cation represented in (A.22),
A
11
A
12
A
21
A
22
B
11
B
12
B
21
B
22
B
A
¼
B
11
A
11
þ
B
12
A
21
B
11
A
12
þ
B
12
A
22
¼
;
(A.24)
B
21
A
11
þ
B
22
A
21
B
21
A
12
þ
B
22
A
22
is an illustration of the fact that
A
A
, the matrices
A
and
B
are said to commute. Finally, matrix multiplication is associative,
B
6¼
B
A
, in general. If
A
B
¼
B
A
ð
B
C
Þ¼ð
A
B
Þ
C
;
(A.25)
and matrix multiplication is distributive with respect to addition
A
ð
B
þ
C
Þ¼
A
B
þ
A
C
and
ð
B
þ
C
Þ
A
¼
B
A
þ
C
A
;
(A.26)
provided the results of these operations are defined.
Example A.3.3
Construct the products
A
B
and
B
A
of the matrices
A
and
B
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
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