Digital Signal Processing Reference
In-Depth Information
20.11.
Derivatives of exponentials of real matrices
. Given a square matrix
X
,the
exponential matrix
e
X
is defined as
e
X
=
I
+
X
+
X
2
2!
+
X
3
3!
+
...
(P20
.
11
a
)
By using the result of Problem 20.9 show that
∂
Tr (
e
X
)
∂
X
=
e
X
T
.
(P20
.
11
b
)
20.12.
Derivatives of exponentials of complex matrices
. For the case where
Z
is
complex show that
∂
Tr (
e
Z
)
∂
Z
∂
Tr (
e
Z
)
∂
Z
∗
Z
T
=
e
and
=
0
(P20
.
12
a
)
for unstructured
Z
,
and
∂
Tr (
e
Z
)
∂
Z
∂
Tr (
e
Z
)
∂
Z
∗
Z
∗
Z
=
e
and
=
e
(P20
.
12
b
)
for Hermitian
Z
.
20.13.
Mimimum or maximum?
Consider again the optimization problem in Ex.
20.16, where the optimum vector
z
and the objective function were shown
to be as in Eqs. (20.47) and (20.48), respectively. Let
b
be any vector
orthogonal to
a
,thatis,
b
†
a
=
0
.
Then the vector
z
=
z
opt
+
b
still satisfies the constraint
z
†
a
=1
.
Show that when
z
opt
is replaced with
z
,
the objective function
φ
opt
changes to
φ
=
φ
opt
+
b
†
Rb
,
where
R
is the Hermitian positive definite matrix given in the optimiza-
tion problem formulation. Based on this, argue that Eq. (20.48) indeed
represents a minimum rather than a maximum.
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