In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.^[1][2]

Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.^[3][2] For example, the 3x4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2x3 block, the top-right 2x1 block, the bottom-left 1x3 block, and the bottom-right 1x1 block.

${\displaystyle \left}$

Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.

This notion can be made more precise for an ${\displaystyle n}$ by ${\displaystyle m}$ matrix ${\displaystyle M}$ by partitioning ${\displaystyle n}$ into a collection ${\displaystyle {\text{rowgroups}}}$, and then partitioning ${\displaystyle m}$ into a collection ${\displaystyle {\text{colgroups}}}$. The original matrix is then considered as the "total" of these groups, in the sense that the ${\displaystyle (i,j)}$ entry of the original matrix corresponds in a 1-to-1 way with some ${\displaystyle (s,t)}$ offset entry of some ${\displaystyle (x,y)}$, where ${\displaystyle x\in {\text{rowgroups}}}$ and ${\displaystyle y\in {\text{colgroups}}}$.^[4]

Block matrix algebra arises in general from biproducts in categories of matrices.^[5]

Example

The matrix

${\displaystyle \mathbf {P} ={\begin{bmatrix}1&2&2&7\\1&5&6&2\\3&3&4&5\\3&3&6&7\end{bmatrix}}}$

can be visualized as divided into four blocks, as

${\displaystyle \mathbf {P} =\left}$.

The horizontal and vertical lines have no special mathematical meaning,^[6][7] but are a common way to visualize a partition.^[6][7] By this partition, ${\displaystyle P}$ is partitioned into four 2×2 blocks, as