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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
then we can define the matrices
and write $L$ as
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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
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.
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