Graphics Reference
In-Depth Information
Here is a diagonal matrix with its determinant
⎡
⎤
200
030
004
⎣
⎦
A
=
|
|=
×
×
=
A
2
3
4
24
.
The identity matrix
I
is a diagonal matrix with a determinant of 1.
4.13 Trace
The
trace
of a square matrix
A
is the sum of its diagonal elements and written as
Tr
(
A
)
. For example:
⎡
⎣
⎤
1234
2345
3456
4567
⎦
A
=
Tr
(
A
)
=
1
+
3
+
5
+
7
=
16
.
In Chap. 9 we use the trace of a square matrix to reveal the angle of rotation
associated with a rotation matrix. And as we will be using the product of two or
more rotation transforms we require to establish that
Tr
(
AB
)
=
Tr
(
BA
)
to reassure ourselves that the trace operation is not sensitive to transform order, and
is readily proved as follows.
Given two square matrices
A
and
B
:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
a
11
···
···
a
1
n
b
11
···
···
b
1
n
···
a
22
···
a
2
n
···
b
22
···
b
2
n
A
=
,
B
=
···
···
···
···
···
···
···
···
a
n
1
···
···
a
nn
b
n
1
···
···
b
nn
then,
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
a
11
···
···
a
1
n
b
11
···
···
b
1
n
···
a
22
···
a
2
n
···
b
22
···
b
2
n
AB
=
···
···
···
···
···
···
···
···
a
n
1
···
···
a
nn
b
n
1
···
···
b
nn
⎡
⎣
⎤
⎦
a
11
b
11
···
···
a
1
n
···
a
22
b
22
···
a
2
n
AB
=
···
···
···
···
a
n
1
···
···
a
nn
b
nn
and Tr
(
AB
)
=
a
11
b
11
+
a
22
b
22
+···+
a
nn
b
nn
.
