R.B. Morris, Y. Tsuji, et al.
International Journal for Numerical Methods in 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.
R.B. Morris, Y. Tsuji, et al.
International Journal for Numerical Methods in Engineering
John R. Kender, Rick Kjeldsen
IEEE Transactions on Pattern Analysis and Machine Intelligence
David W. Jacobs, Daphna Weinshall, et al.
IEEE Transactions on Pattern Analysis and Machine Intelligence
Ziv Bar-Yossef, T.S. Jayram, et al.
Journal of Computer and System Sciences