\begin{definition}[Definition:Block Matrix]
A '''block matrix''' is a representation of a matrix as an array of other matrices.
This may be defined as follows:
Let $S$ be a set.
Let $m, n \ge 1$ be positive integers.
Let $A = \sqbrk {A_{i j} }$ be an $m \times n$ matrix of matrices over $S$.
Let for every $i \in \set {1, \ldots, m}$, the elements of the $i$th row of $A$ have equal height $m_i$.
Let for every $j \in \set {1,\ldots, n}$ the elements of the $j$th column of $A$ have equal width $n_i$.
Define $M = \ds \sum_{i \mathop = 1}^m m_i$ and $N = \ds \sum_{i \mathop = 1}^n n_i$ as indexed summations.
Let more generally $M_k = \ds \sum_{i \mathop = 1}^k m_i$ and $N_l = \ds \sum_{i \mathop = 1}^l n_i$ for $k \in \set {0, \ldots, m}$ and $l \in \set {0, \ldots, n}$.
Then the '''block matrix''' of $A$ is the $M \times N$ matrix $\sqbrk {b_{i j} }$ over $S$ defined as the union of the mappings:
:$b_{i j} = \sqbrk {A_{kl} }_{i - M_{k - 1}, j - N_{l - 1} }$ on $\set {M_{k - 1}, \ldots, M_k} \times \set {N_{l - 1}, \ldots, N_l}$
for $k \in \set {1, \ldots, m}$ and $ l \in \set {1, \ldots, n}$.
Informally, a matrix of matrices $A = \sqbrk {A_{i j} }$ defines a '''block matrix''' by putting together its elements into one big matrix.
It is clear that the orders of the component matrices must be compatible for this construct to be defined.
\end{definition}