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Result A2.3.
For a real-valued function f , the following three conditions are equivalent:
⇔
∂
f
∂
x
=
⇔
∂
f
∂
x
∗
=
∇
x
f
=
0
0
0
.
(A2.28)
In the engineering literature, the gradient of a real-valued function is frequently defined
as (
x
∗
. This definition is justified only insofar
as it can be used to search for local extrema of
f
, thanks to Result
A2.3
.However,
the preceding development has made it clear that (
∂/∂
u
+
j
∂/∂
v
)
f
, which is equal to 2
∂
f
/∂
v
)
f
is
not
the right
definition for the complex gradient of a real (and therefore non-holomorphic) function
of a complex vector
x
.
∂/∂
u
+
j
∂/∂
A2.2
Special cases
In this section, we present some formulae for derivatives of common expressions involv-
ing linear transformations, quadratic forms, traces, and determinants. We emphasize
that there is no need to develop new differentiation rules for Wirtinger derivatives. All
rules for taking derivatives of real functions remain valid. However, care must be taken
to properly distinguish between the variables with respect to which differentiation is
performed and those that are formally regarded as constants.
Some expressions that involve derivatives with respect to a complex vector
x
follow
(it is assumed that
a
and
A
are independent of
x
and
x
∗
):
∂
∂
∂
x
a
H
x
a
H
a
H
x
=
and
=
0
,
(A2.29)
∂
x
∗
∂
∂
∂
x
x
H
a
x
H
a
a
T
=
0
and
=
,
(A2.30)
∂
x
∗
∂
∂
∂
x
x
H
Ax
x
H
A
x
H
Ax
x
T
A
T
=
and
=
,
(A2.31)
∂
x
∗
∂
∂
∂
x
x
T
Ax
x
T
(
A
A
T
)
x
T
Ax
=
+
and
=
0
,
(A2.32)
∂
x
∗
∂
x
exp
−
2
x
H
A
−
1
x
=−
2
exp
−
2
x
H
A
−
1
x
x
H
A
−
1
∂
1
1
1
,
(A2.33)
x
ln
x
H
Ax
=
x
H
Ax
−
1
x
H
A
∂
∂
.
(A2.34)
Sometimes we encounter derivatives of a scalar-valued function
f
(
X
) with respect to
an
n
×
m
complex matrix
X
. These derivatives are defined as
*
,
-
/
*
,
-
/
∂
f
∂
f
∂
f
∂
f
x
11
···
x
11
···
x
1
m
∂
∂
∂
∂
x
1
m
∂
f
∂
f
.
.
.
.
.
.
.
.
.
.
X
=
and
X
∗
=
.
(A2.35)
∂
∂
∂
f
∂
f
∂
f
∂
f
x
n
1
···
x
n
1
···
∂
∂
x
nm
∂
∂
x
nm
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