I have encountered the following problem:
I have two $N$-by-$N$ complex valued matrices $A, B$, and then I form a third matrix as a Hamadard product of the previous two: $C = A \odot B$, so that for each element we have $C_{i,j} = A_{i,j} \cdot B_{i,j}$. My question is whether it is possible to express matrix $C$ as a regular matrix multiplication (dot products) of $A, B$, with a finite number of additional matrices? For instance $C = T^{(L)} \cdot A \cdot T^{(M)} \cdot B \cdot T^{(R)}$, where $T$ are just some auxiliary matrices.
I know that some people have asked this question already, however, the common solutions involve representing matrices in a vector form or using things like SVD. All these as well as using the sum of multiplications on projector operators I want to avoid as my final goal is to find $C^{-1}$, which is why I restricted the desired form of $C$ in terms of dot products.
In general this is impossible. The reason is very simple.
Notice that if $C$ is a product of matrices and $A,B$ are part of this product then $\det(C)=x\det(A)\det(B)$.
Now, choose $A=\begin{pmatrix}1 & 2\\ 1 & 2\end{pmatrix}$ and $B=\begin{pmatrix}1 & 1\\ 1 & -1\end{pmatrix}$ then $C=\begin{pmatrix}1 & 2\\ 1 & -2\end{pmatrix}$ and $\det(C)=-4$, but $\det(A)=0$.
So $\det(C)\neq x\det(A)\det(B)$.