Derivative of a matrix AXBX

Question

I have to find the derivative of a following matrix AXBX, like $$\frac{d(AXBX)}{d(X)}$$ by using matrix differentsation properties where A and B are constant matrices. But I have no idea where to start. Thnx for any help.

Answer 1

Let $f(X) = AXBX$.

$f(X+H)-f(X) = A (X+H)B(X+H) - AXBX = AXBH + AHBX + AHBH$, so we see that $Df(X)(H) = AXBH + AHBX$.

(Since $\|f(X+H)-f(X) - (AXBH + AHBX)\| \le K \|H\|^2$ for some $K$.)

Answer 2

Let $f(X)=AXBX$ and consider $f(X+H)$. We have

$$f(X+H)=A(X+H)B(X+H) = AXBX + AHBX + AXBH + AHBH.$$

In other words, we have

$$f(X+H) = f(X) + \big(AHBX + AXBH\big) + AHBH$$

Show that $Df_X: H\mapsto AHBX + AXBH\,$ is linear, and that $T: H\mapsto AHBH\,$ satisfies

$$\lim_{H\to 0} \,\frac{\lVert T(H)\rVert}{\lVert H\rVert} = 0.$$

Can you see how this solves our problem?

$($Hint for the limit: consider using a submultiplicative norm, that is, one that satisfies $\lVert AB\rVert \leq$ $\lVert A\rVert \,\lVert B\rVert$. Examples include the Frobenius or operator norms.$)$