Digital Signal Processing Reference
In-Depth Information
where # stands for the Moore Penrose inverse. Prove that if the constraint
kuk
2
¼
1 is replaced by
u
0
u ¼
1, the differential
du
given by (4.88) remains valid.
Now consider the same problem where
C
0
is a Hermitian matrix. To fix the
perturbed eigenvector
u
, the condition
kuk
2
¼
1 is not sufficient. So suppose
now that
u
0
u ¼
1. Note that in this case
u
no longer has unit 2-norm. Using
the same approach as for the real symmetric case, prove that the functions
l
(.)
and
u
(.) are differentiable on some neighborhood of
C
0
and that the differentials
at
C
0
are now given by
dl¼ u
0
(
dC
)
u
0
du ¼
(
Cl
0
I
n
)
#
(
I
n
u
0
u
0
)(
dC
)
u
0
:
and
(4
:
89)
In practice, different constraints are used to fix
u
. For example, the SVD function
of MATLAB forces all eigenvectors to be unit 2-norm with a real first element.
Specify in this case the new expression of the differential
du
given by (4.89).
Finally, show that the differential
du
given by (4.88) would be obtained
with the condition
u
0
du ¼
0, which is no longer derived from the constraint
kuk
2
¼
1.
4.2
Consider an
n n
real symmetric or complex Hermitian matrix
C
0
whose the
r
smallest eigenvalues are equal to
s
2
with
l
n
2
r
.
l
n
2
rþ
1
. Let
P
0
the projection
matrix onto the invariant subspace associated with
s
2
. Then a matrix-valued
function
P
(.) is defined as the projection matrix onto the invariant subspace
associated with the
r
smallest eigenvalues of
C
for all
C
in some neighborhood
of
C
0
such that
P
(
C
0
)
¼ P
0
. Using simple perturbations algebra manipulations,
prove that the functions
P
(.) is two times differentiable on some neighborhood
of
C
0
and that the differentials at
C
0
are given by
dP¼
(
P
0
(
dC
)
S
0
þS
0
(
dC
)
P
0
)
þS
0
(
dC
)
P
0
(
dC
)
S
0
P
0
(
dC
)
S
#
0
(
dC
)
P
0
þS
0
(
dC
)
S
0
(
dC
)
P
0
þP
0
(
dC
)
S
0
(
dC
)
S
0
S
#
2
(
dC
)
P
0
(
dC
)
P
0
P
0
(
dC
)
P
0
(
dC
)
S
#
2
0
0
def
C
0
s
2
I
n
.
4.3
Consider a Hermitian matrix
C
whose real and imaginary parts are denoted by
C
r
and
C
i
respectively. Prove that each eigenvalue eigenvector pair (
l
,
u
)of
C
is
associated with the eigenvalue eigenvector pairs
l
,
where
S
0
¼
and
l
,
u
i
u
r
u
r
u
i
where
u
r
and
u
i
denote the real and
C
r
C
i
C
i
of the real symmetric matrix
C
r
imaginary parts of
u
.
4.4
Consider what happens when the orthogonal iteration method (4.11) is
applied with
r ¼ n
and under the assumption that all the eigenvalues of
C
are
simple. The QR algorithm arises by considering how to compute the matrix
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