Define $v_i\in\mathbb{R}^d$ as a row vector, for $i=1,2,...,n$, and $v\in\mathbb{R}^{nd}$ is a row vector concatenated by $v_i$, i.e. $$ v = [v_1, v_2, ..., v_n] $$ Suppose we also have a matrix $M\in\mathbb{R}^{nd\times m}$, concatenated by $n$ matrices $M_i\in\mathbb{R}^{d\times m}$, along the first dimension.
We define a multiplication operator $\odot$ as follow: $$ v\odot M=[v_1*M_1; v_2*M_2;...;v_n*M_n] $$ where $*$ is the matric product operator so that $v_i*M_i\in \mathbb{R}^{1\times m}$; $[ ; ; ]$ is concatenate operator (along the first dimension), so that $v\odot M\in\mathbb{R}^{n\times m}$.
My question is that, whether there is a terminology for operator $\odot$? (I guess maybe it should be called something like "block-wise multiplication"?)
$$\rm v \odot M := \begin{bmatrix} \mathrm v_1^\top & & \\ & \mathrm v_2^\top & & \\ & & \ddots & \\ & & & \mathrm v_n^\top\end{bmatrix} \begin{bmatrix} \mathrm M_1 \\ \mathrm M_2 \\ \vdots \\ \mathrm M_n \end{bmatrix}$$