Graphics Reference
In-Depth Information
2
4
3
5
2
4
3
5
A
11
A
12
A
13
B
11
B
12
B
13
AB ¼
A
21
A
22
A
23
B
21
B
22
B
23
A
31
A
32
A
33
B
31
B
32
B
33
(B.4)
2
4
3
5
A
11
B
11
þ A
12
B
21
þ A
13
B
31
... ...
¼
...
... ...
...
... ...
The
transpose
of a vector or matrix is the original vector or matrix with its rows and columns
exchanged (e.g.,
Eq. B.5
). The
identity matrix
is a square matrix with ones along its diagonal and zeros
3 matrix is formed as shown in
Equation B.8
.
The determinant of matrices greater than 3
3 can be defined recursively. First, define
an element's
submatrix
as the matrix formed when removing the element's row and column from the
original matrix. The determinant is formed by considering any row, element by element. The determi-
nant is the first element of the row times the determinant of its submatrix, minus the second element of
the row times the determinant of its submatrix, plus the third element of the row times the determinant
of its submatrix, and so on. The sum is formed for the entire row, alternating additions and
subtractions.
2
4
3
5
"
#
T
ad
be
cf
abc
def
¼
(B.5)
2
3
100
010
001
4
5
I ¼
(B.6)
MM
1
¼ M
1
M ¼ I
(B.7)
¼ aðei fhÞbðdi fgÞþcðdh egÞ
abc
def
ghi
(B.8)
B.1.1
Inverse matrix and solving linear systems
The inverse of a matrix is useful in computer graphics to represent the inverse of a transformation and in
computer animation to solve a set of linear equations. There are various ways to compute the inverse.
One common method, which is also useful for solving sets of linear equations, is
LU
decomposition.
The basic idea is that a square matrix, for example, a 4
4, can be decomposed into a lower triangular
matrix times an upper triangular matrix. How this is done is discussed later in this section. For now, it is
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