# Definition:Block Matrix

## Informal definition

Informally, a matrix of matrices $A = \sqbrk {A_{i j} }$ defines a block matrix by putting together its elements into one big matrix.

## Definition

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 $M = \displaystyle \sum_{i \mathop = 1}^m m_i$ 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 $k \in \set {0, \ldots, m}$ 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 $l \in \set {1, \ldots, n}$.