Notation for "individual block multiplication" of matrices? 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) ?

$
\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
$$\eqalign{
A\star B
 &= \sum_{k=1}^n \,e_k^T\otimes A_kB_k \\
 &= \sum_{k=1}^n \,e_k^T\otimes{A{E_kE_k^T}B} \\
 &= \sum_{k=1}^n \,e_k^T\otimes{A\LR{e_ke_k^T\otimes I_p}B} \\
}$$
