A note on invariance of the eigenvalues, singular values, and norms of matrix products involving generalized inverses (Q1184479)

It is shown that the set of singular values of the matrix product \(AB^ -C\), as well as any norm of this product, is invariant with respect to the choice of generalized inverse \(B^ -\) of \(B\), if and only if \(AB^ -C\) is invariant with respect to the choice of generalized inverse. In contrast, invariance of the eigenvalues of \(AB^ -C\) with respect to the choice of generalized inverse does not require invariance of \(AB^ -C\).