Global Positioning System Reference
In-Depth Information
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
The minor can be computed for each element of the matrix. It is equal to the
determinant after the respective row and column have been deleted. For example,
the minor for i
=
1 and j
=
2is
a 21
a 23
···
a 2 u
a 31
a 33
···
a 3 u
m 12 =
(A.35)
.
.
.
···
a u 1
a u 3
···
a uu
The cofactor c ij is equal to plus or minus the minor, depending on the subscripts i
and j ,
1 ) i + j
[34
c ij
=
(
m ij
(A.36)
The determinant of A can now be expressed as
Lin
0.8
——
Sho
PgE
u
|
A
|=
a kj c kj
(A.37)
j
=
1
The subscript k is fixed in (A.37) but can be any value between 1 and u ; i.e., the deter-
minant can be computed based on the minors for any one of the u rows or columns.
Of course, the determinant (A.35) can be expressed as a function of determinants of
matrixes of size (u
2 ) , etc.
Determinants have many useful properties. For example, the rank of a matrix
equals the order of the largest nonsingular square submatrix, i.e., the largest order
for a nonzero determinant that can be found. The determinant is zero and the matrix
is singular if the columns or rows of A are linearly dependent. The inverse of the
square matrix can be expressed as
2 )
×
(u
[34
1
A 1
C T
=
(A.38)
|
A
|
where C is the cofactor matrix consisting of the elements c ij given in (A.36). The
product of the matrix and its inverse equals the identity matrix, i.e., AA 1
=
I and
A 1 A
I . These simple relations do not hold for generalized matrix inverses that
can be computed for singular or even rectangular matrices. Information on gener-
alized inverses is available in the standard mathematical literature. The inverse of a
nonsingular square matrix A , B , C , follows the simple rules
=
( ABC ) 1
C 1 B 1 A 1
=
(A.39)
Computation techniques for inverting nonsingular square matrices abound in lin-
ear algebra textbooks. In many cases the matrices to be inverted show a definite pat-
 
Search WWH ::




Custom Search