Global Positioning System Reference
In-Depth Information
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A.3.3 Diagonalization
Consider again a
u
u
matrix
A
and the respective matrix
E
that consists of the
no
rmalized eigenvectors. The product of these two matrices is
×
=
···
AE
[
Ae
1
Ae
2
Ae
u
]
=
λ
1
e
1
λ
2
e
2
···
λ
[
u
e
u
]
(A.46)
=
E
Λ
where
Λ
is a diagonal matrix with
λ
i
as elements at the diagonal. Multiplying this
equation by
E
T
from the left and making use of Equation (A.45), one gets
E
T
AE
=
Λ
(A.47)
[34
Taking the inverse of both sides by applying the rule (A.39) and using (A.45) gives
Lin
—
-0.
——
Nor
*PgE
E
T
A
−
1
E
=
Λ
−
1
(A.48)
Equation (A.47) simply states that if a matrix
A
is premultiplied by
E
T
and postmulti-
plied by
E
, where the columns of
E
are the normalized eigenvectors, then the product
is a diagonal matrix whose diagonal elements are the eigenvalues of
A
. Equation
(A.47) is further modified by
Λ
−
1
/
2
E
T
AE
Λ
−
1
/
2
=
I
(A.49)
[34
Defining the matrix
D
as
Λ
−
1
/
2
D
≡
E
(A.50)
then
D
T
AD
=
I
(A.51)
If the
u
×
u
matrix
A
is positive-semidefinite with rank
R(
A
)
=
r<u
, an equation
similar to (A.47) can be found. Consider the matrix
=
u
F
ru
G
u
−
r
E
(A.52)
where the column of
F
consists of the normalized eigenvectors that pertain to the
r
nonzero eigenvalues. The submatrix
G
consists of
u
−
r
eigenvectors that pertain to
the
u
r
zero eigenvalues. The columns of
F
and
G
span the column and null space,
respectively, of the matrix
A
. Because of Equation (A.40) it follows that
−
AG
=
O
(A.53)
Applying Equations (A.52) and (A.53) gives
