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A partitioned matrix, or a block matrix, is a matrix  that has been constructed from other smaller matrices. These smaller matrices are called blocks or sub-matrices of  .
For instance, if we partition the below  matrix as follows
then we can define the matrices
and write  as
 or  |
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If
 are square matrices (of possibly different sizes), then we define the direct sum of the matrices
 as the partitioned matrix
where the off-diagonal blocks are zero.
If  and  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  and  are  and  matrices, respectively, then if the blocks of  and  are of the correct size to be multiplied, then the blocks of the product are the products of the blocks.
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