Standard notation/operator to "stacking" block matrices?

Question

I looking for some operator or compact notation to stacking vertically and horizontally several matrices (indexed and not indexed). More specifically, i want to represent in a compact way (ie, as an "inline" equation) the following block matrix:

1. $ \begin{bmatrix}A_1 & A_2 & \cdots & A_C\end{bmatrix}$, where $A_i$, $i = 1, \ldots, C$, are matrices with the same number of rows.

2. $ \begin{bmatrix}B_1 \\ B_2 \\ \vdots \\ B_C\end{bmatrix}$, where $B_i$, $i = 1, \ldots, C$, are matrices with the same number of columns.

3. $ \overbrace{\begin{bmatrix}I & \cdots & I\end{bmatrix}}^{\text{C times}}$ and $ \left.\begin{bmatrix}I \\ \vdots \\ I\end{bmatrix}\right\}{\text{C times}}$

where $I$ is an identity matrix.

Is there a standard notation/operator to denote those block matrices?

PS.: I have found the question Notation for juxtaposition operation on matrices, but the answer to that question is not what I looking for.

Answer 1

Using Kronecker products and standard basis vectors $\{e_k\}$ the first two cases are summations $$\eqalign{ A &= \sum_{k=1}^C\,e_k^T\otimes A_k \qquad\qquad B &= \sum_{k=1}^C\,e_k\otimes B_k \\ }$$ By invoking the Einstein summation convention, one could even omit the sigmas.

The third is a special case of these with $\:A_k=I\;\:{\rm or}\;\:B_k=I$