Sign patterns of generalized inverses of M-matrix

Abstract

Let P be an n x n nonnegative stochastic irreducible matrix. Let $A = I - P$ be the associated M-matrix so that A has nonpositive off-diagonal entries and, by the theory of M-matrices, nonnegative principal minors. In this dissertation we investigate under what conditions the group inverse of A remains an M-matrix. For the special classes of stochastic nonnegative matrices comprising of the periodic and nonperiodic Jacobi matrices and of the alternating circulant matrices we determine necessary and sufficient conditions for the group inverse of A to be an M-matrix. Moreover, we investigate the sign patterns of A using a representation formula due to Hunter. Finally, we consider the sign patterns of other generalized inverses of A.