# Definition:Block Matrix

## Definition

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.

## Examples

### Arbitrary Block Matrix: $1$

Let $\mathbf A = \sqbrk a_{m n}, \mathbf B = \sqbrk b_{m p}, \mathbf C = \sqbrk c_{r n}, \mathbf D = \sqbrk d_{r p}$.

The $\paren {m + r} \times \paren {n + p}$ block matrix $\mathbf M$: can then be created as:

$\mathbf M = \begin{bmatrix} \mathbf A & \mathbf B \\ \mathbf C & \mathbf D \end{bmatrix}$

### Arbitrary Block Matrix: $2$

Consider the matrices:

$\mathbf A := \sqbrk \alpha_{m n}$: an $m \times n$ matrix
$\mathbf B := \sqbrk \beta_{p m}$: a $p \times m$ matrix

over an arbitrary ring $R$.

Then:

$\mathbf M = \begin{bmatrix} \mathbf I_m & \mathbf A \\ \mathbf B & \mathbf 0 \end{bmatrix}$

is a matrix whose order is $\paren {m + p} \times \paren {m + n}$ where:

$\mathbf I_m$ is the unit matrix of order $m$
$\mathbf 0$ is the zero matrix of order $p \times n$.

$\mathbf M := \sqbrk \gamma_{\paren {m + p}, \paren {m + n} }$ can be represented as:

$\gamma_{i j} = \begin {cases} \delta_{i j} & : \tuple {i, j} \in \closedint 1 m \times \closedint 1 m \\ \alpha_{i, j - m} & : \tuple {i, j} \in \closedint 1 m \times \closedint {m + 1} {m + n} \\ \beta_{i - m, j} & : \tuple {i, j} \in \closedint {m + 1} {m + p} \times \closedint 1 m \\ 0_R & : \tuple {i, j} \in \closedint {m + 1} {m + p} \times \closedint {m + 1} {m + n} \end {cases}$

## Also see

• Results about block matrices can be found here.