Is there a way of expressing the following expression using only matrix products ?
$(A \circ B ) (A \circ B)^H $
$\circ$ is the Hadamard product and $.^H$ is the conjugate-transpose operator.
$A$ and $B$ are matrices in $\mathbb{C}^{m \times n}$.
hadamard-productlinear algebramatrices
Is there a way of expressing the following expression using only matrix products ?
$(A \circ B ) (A \circ B)^H $
$\circ$ is the Hadamard product and $.^H$ is the conjugate-transpose operator.
$A$ and $B$ are matrices in $\mathbb{C}^{m \times n}$.
Best Answer
According to the comments the only ugly part was the devectorization.
Consider the 3x3 matrix $${\bf X}=\left[\begin{array}{ccc} 1&2&3\\4&5&6\\7&8&9\end{array}\right]$$
It's lexical vectorization is:
$$\text{vec}({\bf X}) = \left[\begin{array}{ccccccccc}1&4&7&2&5&8&3&6&9\end{array}\right]^T$$
We consider that kind of vectorization here.
The Matlab / Octave expression:
evaluates to the zero matrix for several random matrices R, each new term "selecting" a new row.
Then a guess would be that $${\bf R}=\sum_k({\bf v_k} \otimes {\bf I})^T(\text{vec}({\bf R})\otimes {\bf v_k}^T)$$ With $\bf v_k$ being the Dirac or selector vector: $$({\bf v_k})_i = \cases{0, k \neq i\\1, k=i}$$
Feel free to try and simplify it if you want.