Digital Signal Processing Reference
In-Depth Information
where
Q
is an arbitrary
r r
orthogonal matrix. Thus, subspace criterion (4.7) deter-
mines the subspace spanned by
fu
1
,
...
,
u
r
g
or
fu
n
2
rþ
1
,
...
,
u
n
g
, but does not specify
the basis of this subspace at all.
Finally, when now,
l
1
.
l
2
.
.
l
r
.
l
rþ
1
or
l
n
2
r
.
l
n
2
rþ
1
.
.
l
n
2
1
.
l
n
,
3
if (
v
k
)
k¼
1,
... r
denotes
r
arbitrary positive and different real numbers
such that
v
1
.
v
2
.
.
v
r
.
0, the following modification of subspace criterion
(4.7) denoted
weighted subspace criterion
X
r
Tr(
VW
T
CW
)
¼
max
W
T
W¼I
r
v
k
w
k
Cw
k
max
W
T
W¼I
r
k¼
1
or
X
r
Tr(
VW
T
CW
)
¼
min
W
T
W¼I
r
v
k
w
k
Cw
k
min
W
T
W¼I
r
(4
:
8)
k¼
1
with
V ¼
Diag(
v
1
,
...
,
v
r
), has [54]
the unique solution
f+u
1
,
...
,
+u
r
g
or
f+u
n
2
rþ
1
,
...
,
+u
n
g
, respectively.
4.2.4 Standard Subspace Iterative Computational Techniques
The first subspace problem consists in computing the eigenvector associated with
the largest eigenvalue. The
power method
presented in the sequel is the simplest itera-
tive techniques for this task. Under the condition that
l
1
is the unique dominant eigen-
value associated with
u
1
of the real symmetric matrix
C
, and starting from arbitrary
unit 2-norm
w
0
not orthogonal to
u
1
, the following iterations produce a sequence
(
a
i
,
w
i
) that converges to the largest eigenvalue
l
1
and its corresponding eigenvector
unit 2-norm
+u
1
.
w
0
arbitrary such that
w
0
u
1
=
0
for
i ¼
0, 1,
...w
0
iþ
1
¼ Cw
i
w
iþ
1
¼ w
0
iþ
1
=kw
0
iþ
1
k
2
a
iþ
1
¼ w
iþ
1
Cw
iþ
1
:
(4
:
9)
The proof can be found in [35, p. 406], where the definition and the speed of this
convergence are specified in the following. Define
u
i
[
[0,
p
/
2] by cos(
u
i
)
¼
def
jw
i
u
1
j
satisfying cos(
u
0
)
=
0, then
j
sin(
u
i
)
j
tan(
u
0
)
l
2
l
1
i
2
i
and
ja
i
l
1
jjl
1
l
n
j
tan
2
(
u
0
)
l
2
l
1
:
(4
:
10)
Consequently the convergence rate of the power method is exponential and pro-
portional to the ratio
l
1
for the eigenvector and to
l
1
i
2
i
l
2
l
2
for the associated eigenvalue.
If
w
0
is selected randomly, the probability that this vector is orthogonal to
u
1
is equal to
3
Or simply
l
1
.
l
2
.
.
l
n
when
r ¼ n
, if we are interested by all the eigenvectors.
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