Michael E. Henderson
International Journal of Bifurcation and Chaos in Applied Sciences and 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.
Michael E. Henderson
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Ronen Feldman, Martin Charles Golumbic
Ann. Math. Artif. Intell.
I.K. Pour, D.J. Krajnovich, et al.
SPIE Optical Materials for High Average Power Lasers 1992
Shu Tezuka
WSC 1991