Global Positioning System Reference
In-Depth Information
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∂
v
T
Pv
∂
x
=
o
(A.74)
to assure that the least-squares solution for
x
, denoted by
x
,
=−
B
T
PB
−
1
B
T
P
x
(A.75)
φ
at least represents a stationary point of
(
x
)
. In Chapter 4 we verify that indeed a
minimum has also been achieved.
A
.3.5 Matrix Partitioning
Consider the following partitioning of the nonsingular square matrix
N
,
[35
N
11
N
12
N
=
(A.76)
N
21
N
22
Lin
—
0.0
——
Nor
PgE
w
here
N
11
and
N
22
are square matrices, although not necessarily of the same size.
Le
t's denote the inverse matrix by
Q
and partition it accordingly; i.e.,
Q
11
Q
12
N
−
1
Q
=
=
(A.77)
Q
21
Q
22
so
that the sizes of
N
11
and
Q
11
,
N
12
and
Q
12
, etc., are respectively the same. Equa-
tio
ns (A.76) and (A.77) imply the following four relations:
[35
N
11
Q
11
+
N
12
Q
21
=
I
(A.78)
N
11
Q
12
+
N
12
Q
22
=
O
(A.79)
N
21
Q
11
+
N
22
Q
21
=
O
(A.80)
N
21
Q
12
+
N
22
Q
22
=
I
(A.81)
Th
e solutions for the submatrices
Q
ij
are carried out according to the standard rules
fo
r solving a system of linear equations, with the restriction that the inverse is de-
fin
ed only for square submatrices. Multiplying (A.78) from the left by
N
21
N
−
1
11
and
su
btracting the product from (A.80) gives
Q
21
=−
N
22
−
11
N
12
−
1
N
21
N
−
1
N
21
N
−
1
(A.82)
11
Multiplying (A.79) from the left by
N
21
N
−
1
and subtracting the product from (A.81)
11
gives
Q
22
=
N
22
−
11
N
12
−
1
N
21
N
−
1
(A.83)
