Information Technology Reference
In-Depth Information
The
Hessian
matrix of the Rayleigh quotient is given by
4
u
T
ru
uu
T
u
2
ruu
T
u
u
T
ruI
+
2
uu
T
r
u
1
u
H
r
=−
−
r
+
2
+
−
2
2
2
2
2
4
2
C
−
grad
(
r
)
u
T
−
rI
2
u
T
=
−
u
grad
(
r
)
(2.10)
2
2
where
u
is real and
2
u
grad
(
r
)
=
2
(
C
−
rI
)
u
(2.11)
2
It can be observed that
∀
i
=
1, 2,
...
,
n
,
H
r
(
z
i
)
=
C
−
λ
i
I
(2.12)
Hence,
det [
H
r
(
z
i
)
]
=
det [
C
−
λ
i
I
]
=
0
(2.13)
which implies that
H
r
(
z
i
)
is singular
∀
z
i
.Furthermore,
0
i
=
j
λ
j
−
λ
i
z
j
H
r
(
z
i
)
z
j
=
(2.14)
i
=
j
So
H
r
, computed at the RQ critical points, has the same eigenvectors as
C
, but
with different eigenvalues.
H
r
(
u
)
is positive semidefinite only when
u
=
z
min
(see, e.g., [81]).
Proposition 44 (Degeneracy)
The Rayleigh quotient critical points are
degenerate
because in these points the Hessian matrix is not invertible. Then
the Rayleigh quotient is
not
a
Morse function
2
in every open subspace of the
domain containing a critical point.
Remark 45
Among the possible numerical techniques to minimize the Rayleigh
quotient, it is not practical to use the variable metric
(
VM, also known as quasi-
Newton
)
and the Newton descent techniques because the RQ Hessian matrix at
the minimum is singular and then the inverse does not exist. On the other hand,
the conjugate gradient can deal with the singular H
r
[
85, pp. 256-259; 196
]
.
2
A function
f
:
U
→
,
U
being an open subset of
n
, is called a
Morse function
if all its critical
points
x
0
∈
U
are nondegenerate.
Search WWH ::
Custom Search