Definition:Block Matrix

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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.


Sources

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