The Schur complement and the inverse M-matrix problem

It is well known that if M is a nonnegative nonsingular inverse M-matrix and if A is a nonsingular block in the upper left hand corner, then the Schur complement of A in M, (M ? A), is an inverse M-matrix. The converse of this is generally false. In this paper we give added restrictions on M or A to insure the converse, and give some necessary and sufficient conditions for HMH^?1, where H = I⊕(?I), to be an M-matrix.