Digital Signal Processing Reference
In-Depth Information
Note that
D
k
k
k
X
∧
(
A
)(
X
,...,
X
)
=
k
!
(
∧
)
(5.5.3)
and if
k
>
n
or
m
>
k
, then
D
m
k
X
1
X
m
∧
(
A
)(
,...,
)
=
0
.
(5.5.4)
k
In [
8
], Bhatia and Friedland gave the norm of the first derivative of the map
∧
as
follows:
k
D
∧
A
=
p
k
−
1
(
s
1
(
A
),...,
s
k
(
A
)).
The following theorem by Jain [
14
] is an extension of this for its higher order deriv-
atives.
≤
≤
≤
Theorem 5.2
Fo r
1
m
k
n
D
m
k
∧
A
=
m
!
p
k
−
m
(
s
1
(
A
),...,
s
k
(
A
)).
(5.5.5)
n
this reduces to Theorem 2.4 for the determinant map. As a
corollary, a perturbation bound can be obtained using Taylor's theorem.
Note that for
k
=
Corollary 5.3
For any X
∈ M
(
n
)
k
k
k
m
∧
(
A
+
X
)
−∧
(
A
)
≤
p
k
−
m
(
s
1
(
A
), . . . ,
s
k
(
A
))
X
.
(5.5.6)
m
=
1
Consequently,
k
k
k
k
∧
(
A
+
X
)
−∧
(
A
)
≤
(
A
+
X
)
−
A
.
(5.5.7)
5.6 Symmetric Tensor Power
)
→ M
(
n
+
k
−
1
k
)
k
Consider the map
∨
: M
(
n
which takes an
n
×
n
matrix
A
to its
k
th symmetric tensor power. For elements
γ
=
(γ
1
,...,γ
m
)
∈
G
m
,
n
and
G
k
,
n
we write
γ
{
γ
1
,...,γ
m
}⊆
α
=
(α
1
,...,α
k
)
∈
⊆
α
if 1
≤
m
≤
k
and
{
α
1
,...,α
k
}
, with multiplicities allowed such that if
α
occurs in
α
,say
d
α
times,
cannot occur in
γ
times. Also if
γ
⊆
α
, then
α
−
γ
will
then
α
for more than
d
α
denote the element
(γ
1
,...,γ
k
−
m
)
of
G
k
−
m
,
n
, where
γ
∈{
α
1
,...,α
k
}
that is,
γ
and occurs in
α
−
γ
exactly
d
α
−
is some
α
i
d
γ
times where
d
α
and
d
γ
denote the
multiplicities of
α
i
in
α
, and
γ
, respectively.
Let
Y
be a
n
+
m
−
1
m
×
n
+
m
−
1
m
matrix and for 1
≤
m
≤
k
let
γ, δ
∈
G
k
−
m
,
n
.We
,the
n
+
k
−
1
k
×
n
+
k
−
1
k
matrix whose indexing set is
G
k
,
n
, and for
denote by
Y
)
(γ, δ)
(
k
