what is transposition Hadamard product?

Question

$$F=||Q \circ X||_F^2 $$ where $\circ $ is hadamard product. How I can convert it to style of general matrix multiplication?

Answer 1

Let $Q_{ij}$ and $X_{ij}$ denote the elements of $Q$ and $X$.

Then $Q \circ X$ is the matrix whose ijth element = $Q_{ij}X_{ij}$

$F = \Sigma_i\Sigma_j (Q_{ij}X_{ij})^2$

Answer 2

Let's denote by $D_A$ the operation ${\rm Diag}\big({\rm vec}(A)\big)$, i.e. transform the matrix $A$ into a giant diagonal matrix $D_A$.

If you apply this operation to your matrices, you can write the function as $$ F = \|D_X\,D_Q\|_F^2 = {\rm trace}\big(D_X\,D_X\,D_Q\,D_Q\big) $$ where standard matrix multiplications are used.