Global Positioning System Reference
In-Depth Information
F
T
G
T
A
[
FG
]
F
T
AF O
OO
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
Λ
O
OO
E
T
AE
=
=
=
(A.54)
Th
e submatrix contains the
r
nonzero eigenvalues. If
Λ
−
1
/
2
D
=
(
F
G
)
(A.55)
it
follows that
IO
OO
D
T
AD
=
(A.56)
wh
ere the symbol
I
denotes an
r
×
r
identity matrix.
[34
A.3.4 Quadratic Forms
Lin
—
0.6
——
No
PgE
Le
t
A
denote a
u
×
u
matrix and
x
be a
u
×
1 vector. Then
x
T
Ax
v
=
(A.57)
is a quadratic form. The matrix
A
is called positive semidefinite if for all
x
x
T
Ax
≥
0
(A.58)
an
d positive definite if for all
x
[34
x
T
Ax
>
0
(A.59)
Th
e following are some of the properties that are valid for a positive definite matrix
A
:
1.
R(
A
)
u
(full rank).
2.
a
ii
>
0 for all
i
.
3. The inverse
A
−
1
=
is positive definite.
u
matrix with rank
u<n
. Then the matrix
B
T
AB
is positive
4. Let
B
be an
n
×
r<u
, then
B
T
AB
is positive semidefinite.
definite. If
R(
B
)
=
×
−
5. Let
D
be a
q
q
matrix formed by deleting
u
p
rows and the corresponding
−
u
p
columns of
A
. Then
D
is positive definite.
A necessary and sufficient condition for a symmetric matrix to be positive definite is
that the principal minor determinants be positive; i.e.,
a
11
a
12
a
11
>
0
,
>
0
,
...,
|
A
|
>
0
(A.60)
a
21
a
22
or that all eigenvalues are real and positive.
