Graphics Reference
In-Depth Information
Hopefully, it is obvious that reversing the matrix sequence to
BA
only reverses
the
a
and
b
scalar elements on the diagonal, and therefore does not affect the trace
operation.
4.14 Symmetric Matrix
It is worth exploring two types of matrices called
symmetric
and
antisymmetric
ma-
trices, as we refer to them in later chapters. A symmetric matrix is a matrix which
equals its own transpose:
A
T
.
For example, the following matrix is symmetric:
A
=
⎡
⎤
134
324
443
⎣
⎦
.
A
=
The symmetric part of any square matrix can be isolated as follows. Given a matrix
A
and its transpose
A
T
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
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
their sum is
⎡
⎤
2
a
11
a
12
+
a
21
...
a
1
n
+
a
n
1
⎣
⎦
a
12
+
a
21
2
a
22
...
a
2
n
+
a
n
2
A
T
A
+
=
.
.
.
.
.
.
.
a
1
n
+
a
n
1
a
2
n
+
a
n
2
...
2
a
nn
A
T
+
By inspection,
A
is symmetric, and if we divide throughout by 2 we have
2
A
A
T
1
S
=
+
which is defined as the symmetric part of
A
. For example, given
⎡
⎤
⎡
⎤
a
11
a
12
a
13
a
11
a
21
a
31
⎣
⎦
,
A
T
⎣
⎦
A
=
a
21
a
22
a
23
=
a
12
a
22
a
32
a
31
a
32
a
33
a
13
a
23
a
33
then
2
A
A
T
1
S
=
+
⎡
⎤
a
11
(a
12
+
a
21
)/
2
(a
13
+
a
31
)/
2
⎣
⎦
=
(a
12
+
a
21
)/
2
a
22
a
23
+
a
32
(a
13
+
a
31
)/
2
(a
23
+
a
32
)/
2
a
33
