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Concept# Block matrix

Summary

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 by matrix by partitioning into a collection , and then partitioning into a collection . The original matrix is then considered as the "total" of these groups, in the sense that the entry of the original matrix corresponds in a 1-to-1 way with some offset entry of some , where and .
Block matrix algebra arises in general from biproducts in of matrices.
The matrix
can be partitioned into four 2×2 blocks
The partitioned matrix can then be written as
It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions" between two matrices and such that all submatrix products that will be used are defined. Given an matrix with row partitions and column partitions
and a matrix with row partitions and column partitions
that are compatible with the partitions of , the matrix product
can be performed blockwise, yielding as an matrix with row partitions and column partitions. The matrices in the resulting matrix are calculated by multiplying:
Or, using the Einstein notation that implicitly sums over repeated indices:
Helmert–Wolf blocking
If a matrix is partitioned into four blocks, it can be inverted blockwise as follows:
where A and D are square blocks of arbitrary size, and B and C are conformable with them for partitioning. Furthermore, A and the Schur complement of A in P: P/A = D − CA^−1B must be invertible.

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