matrix gradient of the Hadamard product

Question

What is the matrix gradient for the function $||A(B \circ X) ||_F^2 $ with respect to $X$. Here $A,B \in \mathbb{C}^{n \times n}$ and $X \in \mathbb{R}^{n \times n}$.($\circ$) is the Hadamard product and Frobenius norm is used.

Answer 1

Define the matrix $$Y=A(B\circ X)$$ Write the norm in terms of this new matrix and find its differential and gradient $$\eqalign{ \phi &= Y:Y \cr d\phi &= 2Y:dY \cr &= 2Y:A(B\circ dX) \cr &= 2B\circ(A^TY):dX \cr \frac{\partial\phi}{\partial X} &= 2B\circ(A^TY) \cr &= 2B\circ(A^TA(B\circ X)) \cr }$$ In some of the steps, a colon is used to denote the trace/Frobenius product, i.e. $$A:B = {\rm tr}(A^TB)$$