partitioned matrix

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

For instance, if we partition the below $5\\times 5$ matrixas 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 possiblydifferent sizes), then we define the direct sum ofthe 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.