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Block Matrix


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

 [A B; C D],
(1)

where A, B, C, and D are themselves matrices, is a block matrix. In the specific example

A=[0 2; 2 0]
(2)
B=[3 3 3; 3 3 3]
(3)
C=[4 4; 4 4; 4 4]
(4)
D=[5 0 5; 0 5 0; 5 0 5];
(5)

therefore, it is the matrix

 [0 2 3 3 3; 2 0 3 3 3; 4 4 5 0 5; 4 4 0 5 0; 4 4 5 0 5].
(6)

Block matrices can be created using ArrayFlatten.

When two block matrices have the same shape and their diagonal blocks are square matrices, then they multiply similarly to matrix multiplication. For example,

 [A_1 B_1; C_1 D_1][A_2 B_2; C_2 D_2] 
 =[A_1A_2+B_1C_2 A_1B_2+B_1D_2; C_1A_2+D_1C_2 C_1B_2+D_1D_2].
(7)

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 blocks are square n×n matrices, the set of invertible block matrices is a group isomorphic to the general linear group GL_2(R), where R is the ring of square matrices.


See also

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

This entry contributed by Todd Rowland

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Cite this as:

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

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