What is the Hadamard product of matrix $A$ with $BC$, for $B, C$ being matrices?

Question

Say we have three matrices $A,B,C$. Define $\circ$ to be the Hadamard product. The usual matrix product of two matrices $B,C$ is denoted simply as $BC$. Is there an easy expression for $$A\circ(BC)$$?

Answer 1

$\def\D{\operatorname{Diag}}$Let $I$ be the identity matrix. Use the column vectors $(a_k,e_k)$ denote the $k^{th}$ columns of the matrices $(A,I),\,$ respectively. Then one can write the matrix $A$ as a sum of dyadics $$A = \sum_{k=1}^n a_ke_k^T$$ The Hadamard product of a dyad and a matrix obeys $$xy^T\circ M \;=\; \D(x)\cdot M\cdot\D(y)$$ Substituting into the given equation yields $$\eqalign{ A\circ(BC) &= \sum_{k=1}^n a_ke_k\circ(BC) \\ &= \sum_{k=1}^n \D(a_k)\;BC\,\D(e_k) \\ &= \sum_{k=1}^n A_kBCE_k \\ }$$ where $A_k$ is a diagonal matrix constructed from the vector $a_k\,$ (and $E_k$ from $e_k$).