Graphics Programs Reference
In-Depth Information
−
×
−
where
M
ik
is the determinant of the (
n
1)
(
n
1) matrix obtainedbydeleting the
1)
k
+
i
M
ik
iscalleda
cofactor
of
A
ik
.
Equation (A17) isknown as
Laplace's development
of the determinanton the
first row of
A
.Actually Laplace's developmentcan take place on any convenient row.
Choosing the
i
th row, wehave
i
th row and
k
th column of
A
. The term (
−
n
1)
k
+
i
A
ik
M
ik
|
| =
−
A
(
(A18)
k
=
1
The matrix
A
issaid to be
singular
if
|
A
| =
0.
Positive Definiteness
An
n
×
n
matrix
A
issaid to be positive definite if
x
T
Ax
>
0
(A19)
forall nonvanishing vectors
x
. It can be shown that amatrix is positive definite if the
determinants of all its leading minors are positive. The leading minorsof
A
are the
n
square matrices
⎡
⎣
⎤
⎦
A
11
A
12
···
A
1
k
A
12
A
22
···
A
2
k
,
k
=
1
,
2
,...,
n
.
.
.
.
.
.
A
k
1
A
k
2
···
A
kk
Therefore, positive definiteness requires that
>
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
A
11
A
12
A
21
A
22
A
11
>
0
,
0
,
>
0
,...,
|
A
|
>
0
(A20)
Useful Theorems
Welist without proof a few theoremsthat are utilizedinthe main body of the text.
Most proofs areeasy and couldbe attemptedasexercises in matrix algebra.
(
AB
)
T
B
T
A
T
=
(A21a)
(
AB
)
−
1
B
−
1
A
−
1
=
(A21b)
A
T
= |
A
|
(A21c)
|
AB
| = |
A
||
B
|
(A21d)
A
T
BA
where
B
B
T
, then
C
C
T
if
C
=
=
=
(A21e)
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