# 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 PartitionedMatrix 2013-03-22 13:32:55 2013-03-22 13:32:55 mathcam (2727) mathcam (2727) 11 mathcam (2727) Definition msc 15-00 JordanCanonicalForm JordanCanonicalFormTheorem block matrix sub-matrix submatrix