Notation for "individual block multiplication" of matrices?

Question

Suppose I have two block matrices, $A=\begin{pmatrix}A_1 &\cdots&A_n\end{pmatrix}\in\mathbb{R}^{I\times J}$ and $B=\begin{pmatrix}B_1\\\vdots\\B_n\end{pmatrix}\in\mathbb{R}^{J\times I}$, both equally partitioned. Is there a notation or terminology associated to the operation $$ A\,?\,B=\begin{pmatrix}A_1B_1&\ldots&A_nB_n\end{pmatrix} $$ ($A_iB_i$ is a standard matrix multiplication) ?

Answer 1

$ \def\R#1{{\mathbb R}^{#1}} \def\LR#1{\left(#1\right)} $Let $\{e_k\}$ denote the standard basis vectors for $\R{n}$ and $(\otimes)$ the Kronecker product. Let's also write the dimensions of the matrices to make it clear which dimensions are multiples of $n$ $$ A\in\R{m\times np} \qquad A_k\in\R{m\times p} \\ B\in\R{np\times m} \qquad B_k\in\R{p\times m} $$ Use the identity matrix $I_p$ to create block analogs of the standard basis vectors $$\eqalign{ E_k &= \LR{e_k\otimes I_p} \;\in\;\R{np\times p}\\ }$$ which can be used to isolate individual blocks $$\eqalign{ A_k &= AE_k, \qquad B_k &= E_k^TB, \qquad A_kB_k = A\LR{E_kE_k^T}B \\ }$$ Then the proposed product is a sum of such blocks $$\eqalign{ A\star B &= \sum_{k=1}^n \,e_k^T\otimes A_kB_k \\ &= \sum_{k=1}^n \,e_k^T\otimes{A{E_kE_k^T}B} \\ &= \sum_{k=1}^n \,e_k^T\otimes{A\LR{e_ke_k^T\otimes I_p}B} \\ }$$