Global Positioning System Reference
In-Depth Information
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Substituting (A.83) in (A.79) gives
11
N
12
N
22
−
11
N
12
−
1
N
−
1
N
21
N
−
1
Q
12
=−
(A.84)
Substituting (A.82) in (A.78) gives
11
N
12
N
22
−
11
N
12
−
1
N
21
N
−
1
N
−
1
11
N
−
1
N
21
N
−
1
Q
11
=
+
(A.85)
11
An alternative solution for
(
Q
11
,
Q
12
,
Q
21
,
Q
22
)
is readily obtained. Multiplying (A.80)
from the left by
N
12
N
−
1
22
and subtracting the product from (A.78) gives
Q
11
=
N
11
−
22
N
21
−
1
N
12
N
−
1
(A.86)
[35
Substituting (A.86) in (A.80) gives
22
N
21
N
11
−
22
N
21
−
1
N
−
1
N
12
N
−
1
Q
21
=−
(A.87)
Lin
—
0.2
——
No
PgE
Premultiplying (A.81) by
N
12
N
−
1
22
and subtracting (A.79) gives
Q
12
=−
N
11
−
22
N
21
−
1
N
12
N
−
1
N
12
N
−
1
(A.88)
22
Substituting (A.88) in (A.81) gives
22
N
21
N
11
−
22
N
21
−
1
N
12
N
−
1
N
−
1
22
N
−
1
N
12
N
−
1
Q
22
=
+
(A.89)
22
[35
Usually the above partitioning technique is used to reduce the size of large matri-
ces that must be inverted or to derive alternative expressions. Because these matrix
identities are frequently used, and because they look somewhat puzzling unless one
is aware of the simple solutions given above, they are summarized here again to be
able to view them at a glance;
N
11
−
22
N
21
−
1
11
N
12
N
22
−
11
N
12
−
1
N
21
N
−
1
N
12
N
−
1
N
−
1
11
N
−
1
N
21
N
−
1
=
+
(A.90)
11
1
1
N
12
N
22
−
11
N
12
−
1
=
N
11
−
22
N
21
−
1
N
12
N
−
1
N
−
1
N
21
N
−
1
N
12
N
−
1
(A.91)
22
N
22
−
11
N
12
−
1
N
21
N
−
1
22
N
21
N
11
−
22
N
21
−
1
N
21
N
−
1
N
−
1
N
12
N
−
1
=
(A.92)
11
N
22
−
11
N
12
−
1
22
N
21
N
11
−
22
N
21
−
1
N
12
N
−
1
N
21
N
−
1
N
−
1
22
N
−
1
N
12
N
−
1
=
+
(A.93)
22
A.
3.6 Cholesky Factor
For positive definite matrices the square root method, also known as the Cholesky
method, is an efficient way to solve systems of equations and to invert the matrix.
