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

It is shown that if m × n, n × n, and p × m complex matrices A, B, and C do not satisfy at least one of the range inclusions image(A*C*) subset of or equal to image(B*) and image(CA) subset of or equal to image(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 respect to the choice of B− is equivalent to the invariance of the set of all eigenvalues.