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. 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 r o w g r o u p s {displaystyle rowgroups} , and then partitioning m {displaystyle m} into a collection c o l g r o u p s {displaystyle colgroups} . The original matrix is then considered as the 'total' of these groups, in the sense that the ( i , j ) {displaystyle (i,j)} entry of the original matrix corresponds in a 1-to-1 way with some ( s , t ) {displaystyle (s,t)} offset entry of some ( x , y ) {displaystyle (x,y)} , where x ∈ r o w g r o u p s {displaystyle xin {mathit {rowgroups}}} and y ∈ c o l g r o u p s {displaystyle yin {mathit {colgroups}}} . Block matrix algebra arises in general from biproducts in categories of matrices.