partitioned matrix

A partitioned matrix, or a block matrix, is a matrix $M$ that has been constructed from other smaller matrices. These smaller matrices are called blocks or sub-matrices of $M$ .

For instance, if we partition the below $5\times 5$ matrix as follows

$\displaystyle L$

$\displaystyle=$

$\displaystyle\left(\begin{array}[]{cc|ccc}1&0&1&2&3\\ 0&1&1&2&3\\ \hline 2&3&9&9&9\\ 2&3&9&9&9\\ 2&3&9&9&9\\ \end{array}\right),$

then we can define the matrices

$A=\left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right),B=\left(\begin{array}[]{ccc}1&2&3\\ 1&2&3\end{array}\right),C=\left(\begin{array}[]{cc}2&3\\ 2&3\\ 2&3\end{array}\right),D=\left(\begin{array}[]{ccc}9&9&9\\ 9&9&9\\ 9&9&9\\ \end{array}\right)$

and write $L$ as

$L=\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right),\,\mbox{or\,\,}L=\left(\begin{array}[]{c|c}A&B\\ \hline C&D\end{array}\right).$

If $A_{1},\ldots,A_{n}$ are square matrices (of possibly different sizes), then we define the direct sum of the matrices $A_{1},\ldots,A_{n}$ as the partitioned matrix

$\operatorname{diag}(A_{1},\ldots,A_{n})=\left(\begin{array}[]{c|c|c}A_{1}&&\\ \hline&\ddots&\\ \hline&&A_{n}\\ \end{array}\right),$

where the off-diagonal blocks are zero.

If $A$ and $B$ are matrices of the same size partitioned into blocks of the same size, the partition of the sum is the sum of the partitions.

If $A$ and $B$ are $m\times n$ and $n\times k$ matrices, respectively, then if the blocks of $A$ and $B$ are of the correct size to be multiplied, then the blocks of the product are the products of the blocks.

Title	partitioned matrix
Canonical name	PartitionedMatrix
Date of creation	2013-03-22 13:32:55
Last modified on	2013-03-22 13:32:55
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	11
Author	mathcam (2727)
Entry type	Definition
Classification	msc 15-00
Related topic	JordanCanonicalForm
Related topic	JordanCanonicalFormTheorem
Defines	block matrix
Defines	sub-matrix
Defines	submatrix