Further results on invariance of the eigenvalues of matrix products involving generalized inverses

It is shown that if m x n, p x n, and p x m complex matrices A, B, and C do not satisfy at least one of the range inclusions R(A*C*) subset of or equal to R(B*) and R(CA) subset of or equal to R(B), then for each complex number there is a choice of a generalized inverse B- such that this number is an eigenvalue of the product AB(-)C. Combined with criteria derived by Baksalary and Puntanen (1990) and by Baksalary and Pukkila (1992), this result implies that the invariance of the magnitude-largest eigenvalue and/or the magnitude-smallest nonzero eigenvalue of AB(-)C with respect to the choice of B- is equivalent to the invariance of the set of all eigenvalues.