commuting matrices

We consider the properties of commuting matricesMathworldPlanetmath and linear transformations over a vector spaceMathworldPlanetmath V. Two linear transformations φi:VV, i=1,2 are said to commute if for every vV,


If V has finite dimensionPlanetmathPlanetmathPlanetmath n and we fix a basis of V then we may represent the linear transformations as n×n matrices Ai and here the condition of commuting linear transformations is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to testing if their corresponding matrices commute:


Simultaneous triangularisation of commuting matrices over any field can be achieved but may require an extensionPlanetmathPlanetmathPlanetmath of the field. The reason begins to be apparent from the study of eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Remark 1.

Because the implicationMathworldPlanetmath of commuting matrices is best expressed through eigenvectorsMathworldPlanetmathPlanetmathPlanetmath, we prefer the treatment of linear transformations for the .

Recall a linear transformation f:VV is said to leave a subspacePlanetmathPlanetmathPlanetmath EV invariantMathworldPlanetmath if f(E)E.

Proposition 2.

If {φ}iI are commuting linear transformations and E is an eigenspaceMathworldPlanetmath of φi0 for some i0I, then for all iI, φi(E)E.


Let λ be the eigenvalue of φi0 on E. Take any iI and vE. Then


Therefore φi(v)E as E is the λ eigenspace of φi0. In particular, φi(E)E. ∎

We have just shown that commuting linear transformations preserve each other’s eigenspaces. This property does not depend on a finite dimension for V or a finite setMathworldPlanetmath of commuting transformations. However, to characterize commuting linear transformations further will require that V have finite dimension.

Proposition 3.

Let V be a finite dimensional vector space and let {φ}iI be a family of commuting diagonalizablePlanetmathPlanetmath linear transformations from V to V. Then φi can be simultaneously diagonalized.


If a finite dimensional linear transformation is diagonalizable over its field then it has all its eigenvalues in the field (under some basis the matrix is diagonal and the eigenvalues are simply those elements on the diagonal.)

If all the eigenvalues of a linear transformation are the same then the associated diagonal matrixMathworldPlanetmath is scalar. If all φi are scalar then they are simultaneously diagonalized.

Now presume that each φi is not a scalar transformation. Hence there are at least two distinct eigenspaces. It follows each eigenspace of φi has dimension less than that of V.

Now we set up an inductionMathworldPlanetmath on the dimension of V. When the dimension of V is 1, all linear transformations are scalar. Now suppose that for all vector spaces of dimension n, any commuting diagonalizable linear transformations can be simultaneously diagonalized. Then in the case where dimV=n+1, either all the linear transformations are scalar and so simultaneously diagonalized, or at least one is not scalar in which case its eigenspaces are proper subspaces. Since the maps commute they respect each others eigenspaces. So we restrict the maps to any eigenspace and by induction simultaneously diagonalize on this subspace. As the linear transformations are diagonalizable, the sum of the eigenspaces of any φi is V so this process simultaneously diagonalizes each of the φi. ∎

Of course it is possible to have commuting matrices which are not diagonalizable. At the other extreme are unipotent matrices, that is, matrices with all eigenvalues 1. Aside from the identity matrixMathworldPlanetmath, unipotent matrices are never diagonal. Yet they often commute. But here the generalized eigenspacesMathworldPlanetmath substitute for the usual eigenspaces.

It is generally not true that two unipotent matrices commute, even if they share the same eigenspace. For example, the set of unitriangular matrices forms a nilpotent groupMathworldPlanetmath which is abelianMathworldPlanetmathPlanetmath only for 2×2-matrices.

However, if we consider unipotent matrices of the form


we find these to correspond to k×j matrices under addition. Thus this large family of unipotent matrices do commute.

Title commuting matrices
Canonical name CommutingMatrices
Date of creation 2013-03-22 15:54:12
Last modified on 2013-03-22 15:54:12
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 20
Author Algeboy (12884)
Entry type Definition
Classification msc 15A04
Related topic SimultaneousTriangularisationOfCommutingMatricesOverAnyField2
Related topic CommonEigenvectorOfADiagonalElementCrossSection