Frobenius norm derivative

Question

What is the partial derivative of the following expression with respect to T:

$‖X−PTKV‖^2$

Where ∥.∥ denotes the Frobenius norm, and X,P, T , K and V are matrices. thanks.

Answer 1

Let $f(T)=tr((X-PTKV)^T(X-PTKV))$. Then the derivative is

$Df_T:H\rightarrow -2tr((X-PTKV)^TPHKV)=-2tr(KV(X-PTKV)^TPH)=$

$-2<P^T(X-PTKV)V^TK^T,H>$.

Then the gradient is

$\nabla(f)(T)=-2P^T(X-PTKV)V^TK^T$