Graphics Reference
In-Depth Information
⎡
⎤
a
11
s
3
/
2
s
2
/
2
⎣
⎦
=
s
3
/
2
a
22
s
1
/
2
s
2
/
2
s
1
/
2
a
33
where
s
1
=
a
23
+
a
32
s
2
=
a
13
+
a
31
s
3
=
a
12
+
a
21
.
Using a real example:
⎡
⎤
⎡
⎤
014
314
426
034
112
446
⎣
⎦
,
A
T
⎣
⎦
A
=
=
⎡
⎤
024
213
436
⎣
⎦
S
=
which equals its own transpose.
4.15 Antisymmetric Matrix
An
antisymmetric matrix
is a matrix whose transpose is its own negative:
A
T
=−
A
and is also known as a
skew symmetric matrix
.
As the elements of
A
and
A
T
are related by
a
row,col
=−
a
col,row
.
When
k
=
row
=
col
:
a
k,k
which implies that the diagonal elements must be zero. For example, this is an anti-
symmetric matrix
a
k,k
=−
⎡
⎤
06
2
⎣
⎦
.
−
60
−
4
−
24
0
In general, we have
⎡
⎤
⎡
⎤
a
11
a
12
...
a
1
n
a
11
a
21
...
a
n
1
⎣
⎦
⎣
⎦
a
21
a
22
...
a
2
n
a
12
a
22
...
a
n
2
A
T
A
=
,
=
.
.
.
.
.
.
.
.
.
.
.
.
a
n
1
a
n
2
...
a
nn
a
1
n
a
2
n
...
a
nn
