AB is conjugate to BA
If is invertible then . Similarly if is invertible then serves to conjugate to . ∎
The result of course applies to any ring elements and where one is invertible. It also holds for all group elements.
If or is invertible then and have the same eigenvalues.
This leads to an alternate proof of and being almost isospectral. (http://planetmath.org/ABAndBAAreAlmostIsospectral) If and are both non-invertible, then we restrict to the non-zero eigenspaces of so that is invertible on . Thus is conjugate to and so indeed the two transforms have identical non-zero eigenvalues.
|Title||AB is conjugate to BA|
|Date of creation||2013-03-22 16:00:40|
|Last modified on||2013-03-22 16:00:40|
|Last modified by||Algeboy (12884)|