partitioned matrix

A partitioned matrix, or a block matrix, is a matrix M that has been constructed from other smaller matrices. These smaller matrices are called blocks or sub-matrices of M.

For instance, if we partitionMathworldPlanetmathPlanetmath the below 5×5 matrix as follows

L = (1012301123239992399923999),

then we can define the matrices


and write L as

L=(ABCD),or L=(ABCD).

If A1,,An are square matricesMathworldPlanetmath (of possibly different sizes), then we define the direct sumMathworldPlanetmath of the matrices A1,,An as the partitioned matrix


where the off-diagonal blocks are zero.

If A and B are matrices of the same size partitioned into blocks of the same size, the partition of the sum is the sum of the partitions.

If A and B are m×n and n×k matrices, respectively, then if the blocks of A and B are of the correct size to be multiplied, then the blocks of the productPlanetmathPlanetmath are the products of the blocks.

Title partitioned matrix
Canonical name PartitionedMatrix
Date of creation 2013-03-22 13:32:55
Last modified on 2013-03-22 13:32:55
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Definition
Classification msc 15-00
Related topic JordanCanonicalForm
Related topic JordanCanonicalFormTheorem
Defines block matrix
Defines sub-matrix
Defines submatrix