Graphics Reference
In-Depth Information
A matrix is
symmetric
if
A
T
=
A
; that is, if
a
ij
=
a
ji
for all
i
and
j
.If
A
T
=−
A
the
matrix is said to be
skew symmetric
(or
antisymmetric
).
3.1.1
Matrix Arithmetic
Given two
m
B
)
is defined as the pairwise addition of elements from each matrix at corresponding
positions,
c
ij
=
×
n
matrices
A
=[
a
ij
]
and
B
=[
b
ij
]
, matrix
addition
(
C
=[
c
ij
]=
A
+
a
ij
+
b
ij
,or
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
a
11
a
12
···
a
1
n
b
11
b
12
···
b
1
n
a
21
a
22
···
a
2
n
b
21
b
22
···
b
2
n
C
=
A
+
B
=
+
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
···
a
mn
b
m
1
b
m
2
···
b
mn
⎡
⎤
a
11
+
b
11
a
12
+
b
12
···
a
1
n
+
b
1
n
⎣
⎦
a
21
+
b
21
a
22
+
b
22
···
a
2
n
+
b
2
n
=
=[
a
ij
+
b
ij
]
.
.
.
.
.
.
.
a
m
1
+
b
m
1
a
m
2
+
b
m
2
···
a
mn
+
b
mn
Matrix
subtraction
is defined analogously. Multiplication of matrices comes in two
forms. If
c
is a scalar, then the
scalar multiplication
B
=
c
A
is given by
[
b
ij
]=[
ca
ij
]
.
For example:
4
−
−
, where
A
−
.
201
5
8
0
4
201
5
B
=
4
A
=
=
=
−
3
−
1
20
−
12
−
4
−
3
−
1
If
A
is an
m
×
n
matrix and
B
an
n
×
p
matrix, then
matrix multiplication
(
C
=
AB
)
is defined as:
n
=
c
ij
a
ik
b
kj
.
k
=
1
For example:
⎡
⎣
⎤
30
−
.
3
1
47
2
−
17
−
5
⎦
=
C
=
AB
=
2
−
1
−
4
−
6
−
1
−
2
