Matrix differentiation - Signal Processing and Optimization for Transceiver Systems - page 663

The transformation principle. Moon and Stirling [2000] observe that even when
A and B depend on X one can show that

∂ Y
∂x rs

∂y mn
∂ X

= A I B if and only if = A

T

I B

T

rs mn

and

∂ Y
∂x rs

∂y mn
∂ X

= A I

T
rs

B if and only if = B I

T
mn

A .

This is called the transformation principle .

20.2.1 Product rule

Consider the matrix product

Y ( X )= U ( X ) V ( X ) .

Then

= ∂ U
∂x rs

∂ V
∂x rs

,

∂ Y
∂x rs

V + U

where the argument ( X ) has been omitted for brevity. This is the product rule

of matrix calculus, and can be proved easily by writing y mn = k u mk v kn ,and
using the conventional product rule for differentiation.

Example 20.2 .

T

Let Y = X AX . Using the product rule we find

= ∂ X

∂ X
∂x rs

=

T
∂x rs

∂ Y
∂x rs

T T T

AX + X A I

rs AX + X

A I rs .

Example 20.3 .
Let Y = X − 1 .Tocalculate ∂ Y /∂x rs , observe that XX − 1 = I . Differentiating
both sides and using the product rule, we therefore obtain
∂ X
∂x rs

∂ X − 1
∂x rs

= 0 .

X − 1 + X

But since ∂ X /∂x rs =

I rs for arbitrary X , it follows from the above that

∂ X − 1
∂x rs

= − X − 1 I rs X − 1 .

(20 . 5)

If we have an expression like Y = AX − 1 B then using the product rule and
remembering that A and B are constants we get

∂ ( AX − 1 B )
∂x rs

− AX − 1 I rs X − 1 B . =