Further results on invariance of the eigenvalues of matrix product involving generalized inverses (Q1914222)

Let \(L\) be an \(m\times n\) matrix over \(\mathbb{C}\), and let \(L*\) denote its Hermitian conjugate. The generalized inverses of \(L\) are the \(n\times m\) matrices \(L^-\) such that \(LL^- L=L\). Let \(A\), \(B\) and \(C\) denote respectively \(m\times n\), \(p\times n\) and \(p\times m\) matrices over \(\mathbb{C}\). It is known [\textit{J. K. Baksalary} and \textit{T. Pukkila}, Linear Algebra Appl. 165, 125-130 (1992; Zbl 0743.15005)] that the set of eigenvalues of \(AB^- C\) is independent of the choice of \(B^-\) if and only if: (i) \(\text{range} (A* C*) \subseteq \text{range} (B*)\) and \(\text{range} (CA) \subseteq \text{range} (B)\). The main result (Theorem 1) of the present paper is that: if (i) does not hold then for each \(\lambda\in \mathbb{C}\) there is a generalized inverse \(B^-\) such that \(\lambda\) is an eigenvalue of \(AB^- C\).