AB is conjugate to BA

Proposition 1.

Given square matricesMathworldPlanetmath A and B where one is invertiblePlanetmathPlanetmathPlanetmath then AB is conjugatePlanetmathPlanetmath to BA.


If A is invertible then A-1ABA=BA. Similarly if B is invertible then B serves to conjugate BA to AB. ∎

The result of course applies to any ring elements a and b where one is invertible. It also holds for all group elements.

Remark 2.

This is a partial generalizationPlanetmathPlanetmath to the observation that the Cayley table of an abelian groupMathworldPlanetmath is symmetric about the main diagonal. In abelian groups this follows because AB=BA. But in non-abelian groupsMathworldPlanetmath AB is only conjugate to BA. Thus the conjugacy classMathworldPlanetmath of a group are symmetric about the main diagonal.

Corollary 3.

If A or B is invertible then AB and BA have the same eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

This leads to an alternate proof of AB and BA being almost isospectral. (http://planetmath.org/ABAndBAAreAlmostIsospectral) If A and B are both non-invertible, then we restrict to the non-zero eigenspacesMathworldPlanetmath E of A so that A is invertible on E. Thus (AB)|E is conjugate to (BA)|E and so indeed the two transforms have identical non-zero eigenvalues.

Title AB is conjugate to BA
Canonical name ABIsConjugateToBA
Date of creation 2013-03-22 16:00:40
Last modified on 2013-03-22 16:00:40
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 4
Author Algeboy (12884)
Entry type Theorem
Classification msc 15A04