The inverse M-matrix problem

Abstract

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(aij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ≤ aij ≤ x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A-1=B=(bij), then bii> 0 and bij ≤ 0 for i≠j. If n=2 or x=y no further conditions are needed, but if n ≥ 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x2=sy+(1-s)y2; then B is an M-matrix if s-1 ≥ n-2. Moreover, if all off-diagonal elements of A have the value y except for aij=ajj=x when i=n-1, n and 1 ≤ j ≤ n-2, then the condition on both necessary and sufficient for B to be an M-matrix. © 1977.