Proposition 1   Given square matrices and where one is invertible then is conjugate to .

Proof. 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.

Remark 2   This is a partial generalization to the observation that the Cayley table of an abelian group is symmetric about the main diagonal. In abelian groups this follows because . But in non-abelian groups is only conjugate to . Thus the conjugacy class of a group are symmetric about the main diagonal.

Corollary 3   If or is invertible then and have the same eigenvalues.

This leads to an alternate proof of and being almost isospectral. 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.