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

$L$$=$

then we can define the matrices

and write $L$ as

If $A_1,\mathrm{\dots },A_n$ are square matrices (of possiblydifferent sizes), then we define the direct sum ofthe matrices $A_1,\mathrm{\dots },A_n$as the partitioned matrix

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.