Global Positioning System Reference
In-Depth Information
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tern and are often sparsely populated. When solving large systems of equations, it
might be necessary to take advantage of these patterns in order to reduce the com-
putation load (George and Liu, 1981). Very useful subroutines are available in the
public domain, e.g., Milbert (1984). Some applications might produce ill-conditioned
(numerically near-singular) matrices that require special attention.
A.
3.2 Eigenvalues and Eigenvectors
Let
A
denote a
u
×
u
matrix and
x
be a
u
×
1 vector. If
x
fulfills the equation
Ax
= λ
x
(A.40)
λ
it is called an eigenvector, and the scalar
is the corresponding eigenvalue. Equation
[34
(A.40) can be rewritten as
(
A
− λ
I
)
x
=
o
(A.41)
Lin
—
0.2
——
Sho
PgE
If
x
0
denotes a solution of (A.41) and
x
0
is also a solution. It follows
that (A.41) provides only the direction of the eigenvector. There exists a nontrivial
solution for
x
if the determinant is zero; i.e.,
α
is a scalar, then
α
|
A
− λ
I
|=
0
(A.42)
This is the characteristic equation. It is a polynomial of the
u
th order in
λ
, providing
u
solutions
,u
. Some of the eigenvalues can be zero, equal (multiple
solution), or even complex. Equation (A.41) provides an eigenvector
x
i
for each
eigenvalue
λ
i
, with
i
=
1
,
···
[34
λ
i
.
For a symmetric matrix, all eigenvalues are real. Although the characteristic equa-
tion might have multiple solutions, the number of zero eigenvalues is equal to the
rank defect of the matrix. The eigenvectors are mutually orthogonal,
x
i
x
j
=
0
(A.43)
For positive-definite matrices all eigenvalues are positive. Let the normalized eigen-
vectors
x
i
/
x
i
be denoted
e
i
; we can combine the normalized eigenvectors into a
matrix,
E
=
[
e
1
e
2
···
e
u
]
(A.44)
The matrix
E
is an orthonormal matrix for which
E
T
E
−
1
=
(A.45)
holds.
