direct sum of matrices

Direct sum of matrices

Let A be an m×n matrix and B be a p×q matrix. By the direct sumPlanetmathPlanetmathPlanetmath of A and B, written AB, we mean the (m+p)×(n+q) matrix of the form


where the O’s represent zero matricesMathworldPlanetmath. The O on the top right is an m×q matrix, while the O on the bottom left is n×p.

For example, if A=(3-125) and B=(1240-78), then


Remark. It is not hard to see that the operation on matrices is associative:


because both sides lead to


In fact, we can inductively define the direct sum of n matrices unambiguously.

Direct sums of linear transformations

The direct sum of matrices is closely related to the direct sum of vector spacesMathworldPlanetmath and linear transformations. Let A and B be as above, over some field k. We may view A and B as linear transformations TA:knkm and TB:kqkp using the standard ordered bases. Then AB may be viewed as the linear transformation


using the standard ordered basis, such that

  • the restriction of TAB to the subspacePlanetmathPlanetmath kn (embedded in kn+q) is TA, and

  • the restriction of TAB to kq is TB.

The above suggests that we can define direct sums on linear transformations. Let T1:V1W1 and T2:V2W2 be linear transformations, where Vi and Wj are finite dimensional vector spaces over some field k such that V1V2=0. Then define T1T2:V1V2W1W2 such that for any vV1V2,


where viVi. Based on this definition, it is not hard to see that


for any matrices A and B.

More generally, if βi is an ordered basis for Vi, then β:=β1β2 extending the linear orders on βi, such that if viβ1 and vjβ2, then vi<vj is an ordered basis for V1V2, and

Title direct sum of matrices
Canonical name DirectSumOfMatrices
Date of creation 2013-03-22 17:36:48
Last modified on 2013-03-22 17:36:48
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 15-01
Related topic DirectSum