Matrix differentiation - Signal Processing and Optimization for Transceiver Systems - page 669

20.3.1 Definitions

| 2 = zz ∗ this is
not analytic in the variable z (the presence of z ∗ makes it non-analytic [Churchill
and Brown, 1984]). So dh ( z ) /dz does not exist. However, imagine for a moment
that z and z ∗ denote two separate, independent, variables and define the two-
variable function

| 2 . Since To motivate the idea consider the example h ( z )= | z | z

g ( z, z ∗ )= zz ∗ .
We can differentiate with respect to z by holding z ∗ constant, and vice versa.
That is, we can define the partial derivatives
∂g ( z, z ∗ )
∂z

∂g ( z, z ∗ )
∂z ∗

= z ∗

and = z.

A subtle point about notation. We know that z and z ∗ are not independent;
knowing one we can find the other. However, given the original function h ( z ), if
we replace all occurrence of z with the notation z 1 and all occurrence of z ∗ with
z 2 , then the two-variable function g ( z 1 ,z 2 ) can certainly be differentiated with
respect to each variable separately. Instead of using the new notations z 1 and
z 2 we simply carry on with z and z ∗ . The fact that any h ( z ) can be expressed
uniquely as g ( z, z ∗ ) is explained in a footnote in Sec. 20.3.2.

The usefulness of this “brave” definition will become clear as soon as some
of its properties are established. For now, just consider another example: let
h ( z )=2Re[ z ]. This is not analytic because h ( z )= z + z ∗ .So dh ( z ) /dz does not
exist. Define the two-variable function

g ( z, z ∗ )= z + z ∗ .

Then

∂g ( z, z ∗ )
∂z

∂g ( z, z ∗ )
∂z ∗

= 1 and =1 .

Finally consider the simple example h ( z )= z. In this case g ( z, z ∗ )= z so that
∂g/∂z = 1 and ∂g/∂z ∗ =0 . Thus

∂z
∂z = 1

∂z
∂z ∗

and =0 . (20 . 18)

Similarly,

∂z ∗
∂z

∂z ∗
∂z ∗

= 0 and =1 . (20 . 19)

More generally,

if g ( z, z ∗ ) is free from z ∗ then ∂g/∂z ∗ =0 ,

(20 . 20)

if g ( z, z ∗ ) is free from z then ∂g/∂z =0 . (20 . 21)
In complex variable theory the derivative ∂z ∗ /∂z is undefined because z ∗ is not
analytic. Observe however that the meaning of the notation here is different.