M-matrix generalized inverses of M-matrices Original Research Article

Let P be an n × n nonnegative, irreducible, and stochastic matrix, and consider the associated singular and irreducible M-matrix Q = I − P. We show that without further restrictive spectral or combinatorial assumptions on P, it is reasonable to expect that the group inverse Qmusical sharp of Q will be an M-matrix only when P lies in a small wedge about a rank 1 nonnegative matrix, a wedge whose existence is itself dependent on P. We go on to consider the question of when Q has a nonzero generalized inverse, not necessarily the group inverse, which is an M-matrix. We give examples of all sorts of classes of P for which such is the case, but point out that the general problem remains unsolved.