Biomedical Engineering Reference
In-Depth Information
Addition and subtraction of matrices are valid only for matrices of the same order. Either
operation is applied to corresponding elements term by term, that is,
. . .
. . .
a
11
a
12
a
1
m
b
11
b
12
b
1
m
. . .
. . .
a
21
a
22
a
2
m
b
21
b
22
b
2
m
+
. . .
. . .
a
n
1
a
n
2
a
nm
b
n
1
b
n
2
b
nm
ð
A
:
3
Þ
. . .
c
11
=
a
11
+
b
11
c
12
=
a
12
+
b
12
c
1
m
=
a
1
m
+
b
1
m
. . .
c
21
=
a
21
+
b
21
c
22
=
a
22
+
b
22
c
2
m
=
a
2
m
+
b
2
m
=
. . .
c
n
1
=
a
n
1
+
b
n
1
c
n
2
=
a
n
2
+
b
n
2
c
nm
=
a
nm
+
b
nm
where, in general,
.
Matrix multiplication is valid only when the number of columns in the first matrix is
equal to the number of rows in the second matrix. If
A
, order
c
ij
¼
a
ij
þ
b
ij
. Subtraction follows similarly with
c
ij
¼
a
ij
b
ij
n
m
, is multiplied by
B
,
order
m
n
, then the order of the resulting matrix
C
is
n
n
. Each element of
C
,
c
ij
, is equal
th
row of
A
with the
th
column of
B
, that is
to the sum of products of the
i
j
X
n
c
ij
¼
1
a
ik
b
kj
ð
A
:
4
Þ
k
¼
for
. In general, matrix multiplication for matrices of the same
order is not commutative, that is,
AB
i
¼
1, 2,
...
,
n
and
j
¼
1, 2,
...
,
m
BA
. The multiplication of a matrix by a scalar a
equals the product of each element of the matrix by a.
The identity matrix
I
is a square matrix whose nondiagonal elements are zero and whose
diagonal elements are one, that is
6¼
. . .
10 0
0
. . .
01 0
0
. . .
00 1
0
ð
A
:
5
Þ
I
=
. . .
00 0
1
The null matrix,
0
, has 0 for all elements in the matrix.
Matrix operations of addition, subtraction, multiplication, and division follow much the
same processes that these operations do with real numbers. For arbitrary and appropriately
defined matrices
A
,
B
,and
C
, we have
Commutative Property: A
þ
¼
þ
B
B
A
þ
þ
¼
þ
þ
Associative Property: A
(B
C)
(A
B)
C
Distributive Property: A(B
þ
C)
¼
AB
þ
AC
Identities Involving I and 0:
AI
¼
A
0A
¼
0
A0
¼
0
A
þ
0
¼
A
