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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. 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. 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 n {\displaystyle n} by m {\displaystyle m} matrix M {\displaystyle M} by partitioning n {\displaystyle n} into a collection rowgroups {\displaystyle {\text{rowgroups}}} , and then partitioning m {\displaystyle m} into a collection colgroups {\displaystyle {\text{colgroups}}}. The original matrix is then considered as the "total" of these groups, in the sense that the {\displaystyle } entry of the original matrix corresponds in a 1-to-1 way with some {\displaystyle } offset entry of some {\displaystyle } , where x ∈ rowgroups {\displaystyle x\in {\text{rowgroups}}} and y ∈ colgroups {\displaystyle y\in {\text{colgroups}}}.
Block matrix algebra arises in general from biproducts in categories of matrices.