Group inverses of M-matrices associated with nonnegative matrices having few eigenvalues Original Research Article

We continue here earlier investigations of the structure of group generalized inverses A# of singular and irreducible M-matrices A = rI − M, where M is an n × n nonnegative and irreducible matrix whose spectral radius is r and whereM is assumed to belong to one of several special classes of nonnegative matrices. We now focus on cases where M has only a few distinct eigenvalues. We are particularly interested in instances of such matricesM which cause all or almost all of the off-diagonal entries of the corresponding A# to be nonpositive. We then apply our results to the study of the sign pattern of the off-diagonal entries of the group generalized inverse of A = rI − M, where M comes from one of the following families of nonnegative matrices: (1) the magic square of ordern = 4k generated by Matlab, (2) the doubly regular tournament matrices. The latter type is an example of a special type of a nonnegative matrix M for which the group inverse of the associatedM-matrix satisfiesA# = α At for someα > 0.