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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