Digital Signal Processing Reference
In-Depth Information
20.7.
Derivatives of logarithms of determinants
. For a real square matrix
X
,
the
quantity
∂
[ln det(
X
)]
/∂x
rs
can be calculated by using the standard chain
rule of calculus. Using this idea show that
=
X
−T
∂
[ln det(
X
)]
∂
X
for unstructured
X
(P20
.
7)
2
X
−T
diag(
X
−T
)
−
for symmetric
X
.
20.8.
Derivatives of logarithms of determinants, complex case
. By extending the
methods of the preceding problem to complex square matrices
Z
,
show
that
∂
[ln det(
Z
)]
∂
Z
∂
[ln det(
Z
)]
∂
Z
∗
=
Z
−T
and
=
0
(P20
.
8
a
)
if
Z
is unstructured, and
∂
[ln det(
Z
)]
∂
Z
∂
[ln det(
Z
)]
∂
Z
∗
=
Z
−T
=(
Z
−
1
)
∗
=
Z
−
1
and
(P20
.
8
b
)
when
Z
is Hermitian.
20.9.
Derivatives of powers of matrices
. For a real square matrix
X
show that
n−
1
n
∂x
rs
∂
X
k
n−k−
1
.
=
X
I
rs
X
(P20
.
9
a
)
k
=0
For example,
∂
X
2
/∂x
rs
=
I
rs
X
+
X
I
rs
.
From the preceding show that
n
)
∂
Tr (
X
n−
1
)
T
.
=
n
(
X
(P20
.
9
b
)
∂
X
20.10.
Derivatives of powers of complex matrices
. For a square complex matrix
Z
show that
n−
1
n
n
∂z
rs
∂
Z
∂
Z
k
n−k−
1
=
Z
I
rs
Z
and
=0
(P20
.
10
a
)
∂z
rs
k
=0
when
Z
is unstructured. Show also that
n−
1
n−
1
n
n
∂
Z
∂
Z
k
n−k−
1
k
T
n−k−
1
=
Z
I
rs
Z
and
=
Z
I
rs
Z
(P20
.
10
b
)
∂z
rs
∂z
rs
k
=0
k
=0
when
Z
is Hermitian. Taking traces and simplifying show that
n
)
n
)
∂
Tr (
Z
∂
Tr (
Z
n−
1
)
T
=
n
(
Z
and
=
0
(P20
.
10
c
)
∂
Z
∗
∂
Z
for unstructured
Z
,
and
n
)
n
)
∂
Tr (
Z
∂
Tr (
Z
n−
1
)
∗
n−
1
)
=
n
(
Z
and
=
n
(
Z
( 20
.
10
d
)
∂
Z
∂
Z
∗
for Hermitian
Z
.
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