Hang-Yip Liu, Steffen Schulze, et al.
Proceedings of SPIE - The International Society for Optical Engineering
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.
Hang-Yip Liu, Steffen Schulze, et al.
Proceedings of SPIE - The International Society for Optical Engineering
S.F. Fan, W.B. Yun, et al.
Proceedings of SPIE 1989
Imran Nasim, Melanie Weber
SCML 2024
Ronen Feldman, Martin Charles Golumbic
Ann. Math. Artif. Intell.