A note on invariance of the eigenvalues, singular values, and norms of matrix products involving generalized inverses

Abstract

It is shown that the set of singular values of the product AB - C, as well as any norm of this product, is invariant with respect to the choice of a generalized inverse B - if and only if AB - C is invariant itself. This is in contrast with the corresponding invariance property of the set of eigenvalues of ⊙AB -⊙C, for which the invariance of AB - C is not necessary.