Graphics Reference
In-Depth Information
1
a
11
a
22
−
a
22
−
a
12
A
−
1
=
−
a
21
a
11
a
12
a
21
which, for the above matrix is
−
13
4
.
=
−
1
14
1
14
1
−
3
A
−
1
=
−
42
−
2
4.11 Orthogonal Matrix
Although many matrices have to be inverted using the transpose of their cofactor
matrix, an
orthogonal matrix
implies that its transpose is also its inverse. For exam-
ple:
cos
β
−
sin
β
A
=
sin
β
cos
β
is orthogonal because
cos
β
sin
β
A
T
=
−
sin
β
cos
β
and
cos
β
cos
β
10
01
.
−
sin
β
sin
β
AA
T
=
=
sin
β
cos
β
−
sin
β
cos
β
Orthogonal matrices play an important role in rotations because they leave the origin
fixed and preserve all angles and distances. Consequently, an object's geometric
integrity is maintained after a rotation, which is why an orthogonal transform is
known as a
rigid motion
transform.
A rotation transform also preserves orientations, which means that left-handed
and right-handed axial systems (frames) remain unaltered after a rotation. Such
changes in orientation will occur with a reflection transform.
4.12 Diagonal Matrix
A
diagonal matrix
is a square matrix whose elements are zero, apart from its diag-
onal:
⎡
⎣
⎤
⎦
a
11
0
...
0
0
a
22
...
0
A
=
.
.
.
.
.
.
.
00
...
a
nn
The determinant of a diagonal matrix must be
det
A
=
a
11
×
a
22
×···×
a
nn
.
