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A block matrix is a matrix that is defined using smaller matrices, called blocks. For example,

where , , , and are themselves matrices, is a block matrix. In the specific example

therefore, it is the matrix

Block matrices can be created using ArrayFlatten.

When the column-block sizes of the first block matrix match the row-block sizes of the second, block multiplication follows the usual rule for matrix multiplication. For example,

Note that the usual rules of matrix multiplication hold even when the block matrices are not square (assuming that the block sizes correspond). Of course, matrix multiplication is in general not commutative, so in these block matrix multiplications, it is important to keep the correct order of the multiplications.

When the four blocks are square matrices, the set of invertible block matrices is a group isomorphic to the general linear group , where is the ring of square matrices.

See also

Block, Block Diagonal Matrix, Cayley-Hamilton Theorem, Controllability Matrix, Matrix, Ring, Submatrix

This entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Block Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BlockMatrix.html

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