Matrix differentiation - Signal Processing and Optimization for Transceiver Systems - page 672

and

⎡
⎣

⎤
⎦

∂g ( Z , Z ∗ )
∂z 00

∂g ( Z , Z ∗ )
∂z 01

...

∂g ( Z , Z ∗ )
∂ Z ∗

∂g ( Z , Z ∗ )
∂z 10

∂g ( Z , Z ∗ )
∂z 11

=

...

. .

. . .

The properties of complex gradients outlined for the scalar case can be applied
to each element in the matrix or the vector to arrive at the properties of complex
vector gradients and complex matrix gradients. For example, if g ( Z , Z ∗ )does
not have any Z ∗ in it, then

∂g ( Z , Z ∗ )
∂ Z ∗

= 0 .

Also, Eqs. (20.23) can be applied elementwise to obtain

=0 . 5 ∂f ( X , Y )
∂ X

∂g ( Z , Z ∗ )
∂ Z

j ∂f ( X , Y )
∂ Y

− (20 . 26)

and

=0 . 5 ∂f ( X , Y )
∂ X

,

∂g ( Z , Z ∗ )
∂ Z ∗

+ j ∂f ( X , Y )
∂ Y

(20 . 27)

where Z = X + j Y and g ( Z , Z ∗ )= f ( X , Y ) .

20.3.5 Stationary points and optimization

Let g ( Z , Z ∗ )bea real scalar function and write g ( Z , Z ∗ )= f ( X , Y )asusual.
By definition, the stationary point of g is such that

∂f ( X , Y )
∂ X

∂f ( X , Y )
∂ Y

= 0 and = 0 .

Since g (and hence f ) is real, it follows from Eq. (20.26) that a point is a

stationary point if and only if

∂g ( Z , Z ∗ )
∂ Z

= 0 (20 . 28)

at that point. We also see that a point is a stationary point if and only if

∂g ( Z , Z ∗ )
∂ Z ∗

= 0 (20 . 29)

at that point. So either of the above conditions can be used to test stationarity.
We conclude with the following remarks:

1. Equations (20.28) and (20.29) are fundamental in applications involving
optimization .Ata local interior extremum of the real function g ( Z , Z ∗ ) ,
the gradient ∂g ( Z , Z ∗ ) /∂ Z (equivalently ∂g ( Z , Z ∗ ) /∂ Z ∗ ) is set to zero to
obtain a necessary condition for optimality.