# Definition:Block Matrix

## Informal definition

Informally, a matrix of matrices $A = (A_{ij})$ defines a block matrix by putting together its elements into one big matrix.

## Definition

Let $S$ be a set.

Let $m,n \geq 1$ be positive integers.

Let $A = (A_{ij})$ be an $m\times n$ matrix of matrices over $S$.

Let for every $i \in \{1,\ldots, m\}$, the elements of the $i$th row of $A$ have equal height $m_i$.

Let for every $j \in \{1,\ldots, n\}$ the elements of the $j$th column of $A$ have equal width $n_i$.

Define $M= \displaystyle \sum_{i\mathop=1}^m m_i$ and $N=\displaystyle \sum_{i\mathop=1}^n n_i$ as indexed summations.

Let more generally $M_k = \displaystyle \sum_{i\mathop=1}^k m_i$ and $N_l =\displaystyle \sum_{i\mathop=1}^l n_i$ for $k \in \{0, \ldots, m\}$ and $l \in \{0, \ldots, n\}$.

Then the block matrix of $A$ is the $M \times N$ matrix $(b_{ij})$ over $S$ defined as the union of the mappings

$b_{ij} = (A_{kl})_{i-M_{k-1}, j-N_{l-1}}$ on $\{ M_{k-1}, \ldots, M_{k}\} \times \{ N_{l-1}, \ldots, N_{l}\}$

for $k \in \{1, \ldots, m\}$ and $l \in \{1, \ldots, n\}$.