Block matrix

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 ${\displaystyle n}$ by ${\displaystyle m}$ matrix ${\displaystyle M}$ by partitioning ${\displaystyle n}$ into a collection ${\displaystyle rowgroups}$, and then partitioning ${\displaystyle m}$ into a collection ${\displaystyle colgroups}$. 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 ${\displaystyle x\in {\mathit {rowgroups}}}$ and ${\displaystyle y\in {\mathit {colgroups}}}$.

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