Digital Signal Processing Reference
In-Depth Information
is
m
(α
−
γ)
m
(β
−
δ)
m
1
/
2
α, β
∈
G
k
,
n
the
(α, β)
-entry of
Y
)
(γ, δ)
times
(α
−
γ, β
−
δ)
-
(
k
(α)
m
(β)
entry of
Y
if
γ
⊆
α
and
δ
⊆
β
and zero otherwise.
We know that for any
α, β
∈
k
A
is
G
k
,
n
the
(α, β)
-entry of
∨
(β))
−
1
/
2
per
A
k
A
by
(
m
(α)
m
[
α
|
β
]
. Calculating the derivative of each entry of
∨
using the results from Sect.
5.3
will lead to the following.
Theorem 6.1
Let A
∈ M
(
n
)
. Then for
1
≤
m
≤
k
D
m
k
X
1
X
m
X
1
X
m
∨
(
A
)(
,...,
)
=
m
!
per
A
[
γ
|
δ
]
(
∨···∨
)
(
k
)
(γ, δ).
γ,δ
∈
G
k
−
m
,
n
(5.6.1)
We note here that
D
k
k
k
X
∨
(
)(
,...,
)
=
!
(
∨
)
A
X
X
k
(5.6.2)
and for
m
>
k
D
m
k
X
1
X
m
∨
(
A
)(
,...,
)
=
0
.
(5.6.3)
k
:
Bhatia [
6
] computed the exact norm of the first derivative of the map
∨
k
k
−
1
D
∨
(
A
)
=
k
A
.
k
We extend this result for all order derivatives of the map
∨
in [
11
].
Theorem 6.2
Fo r
1
≤
m
≤
k
k
!
D
m
k
k
−
m
∨
A
=
)
!
A
.
(5.6.4)
(
k
−
m
n
A
for
α
=
(
Since per
A
is
(α, α)
-entry of
∨
1
,...,
n
)
, Theorem 3.5 follows from
the above theorem, by putting
k
n
.
By Taylor's theorem, we obtain higher order perturbation bounds.
=
Corollary 6.3
Fo r X
∈ M
(
n
)
k
k
A
k
k
∨
(
+
)
−∨
≤
(
+
)
−
.
A
X
A
X
A
(5.6.5)
5.7 Coefficients of Characteristic Polynomial
The
characteristic polynomial
of
A
is defined by
det
(
xI
−
A
).
