Block Matrix with Two Row Blocks and Two Column Blocks

Let $m$ and $n$ be natural numbers with $1\le m$ and $1\le n$, let $\mathbb{R}$ be the real numbers, and for a natural number $p$ let $[p]$ be the initial segment determined by $p$. Let $A$, $B$, $C$ and $D$ be real matrices of sizes $m\times m$, $m\times n$, $n\times m$ and $n\times n$ respectively.

As recorded in Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space, every $k\in[m+n]$ satisfies exactly one of the following: $k\in[m]$; or $k=m+i$ for a unique $i\in[n]$.

The block matrix

$$\begin{pmatrix}A&B\\C&D\end{pmatrix}$$

is the real $(m+n)\times(m+n)$ matrix $M$ whose entries are given, for $k,l\in[m]$ and $i,j\in[n]$, by

$$M_{kl}=A_{kl},\qquad M_{k,\,m+j}=B_{kj},\qquad M_{m+i,\,l}=C_{il},\qquad M_{m+i,\,m+j}=D_{ij} .$$