Partial Derivative of function with Matrix X^T, X and U.

Question

Trying to wrap my head around something I can't seem to place in the Matrix cookbook.

If I have a function

f = $\mathrm( 2X^TXU)$.

Where X and U are both matrices.

I want to take the derivative of f w.r.t U.

The closest I can find in the matrix cookbook is (113) but that's the opposite case where I have U^T * U * X and am taking derivative w.r.t X.

What is the appropriate course of action in this case?

Thanks!

Answer 1

As for any fixed $X$ the function $$F: U\longmapsto 2X^TXU$$ is linear and continuous, the (Fréchet) derivative of $F$ in any $U_0$ is the same function: $$DF(U_0) = F.$$

Answer 2

As $f$ is linear with respect to $U$, its differential on any point is itself.

One way to prove it is to compute $f(U+H)$:

$$f(U+H)=f(U)+f(H)$$

so $df(U)(H)=f(H)$.