Preferred basis, quasi-preferred basis, height characteristic, level characteristic
An M_∨-matrix has the form A = sI − B with s ≥ ρ(B) and B^k is entrywise nonnegative for all sufficiently large integers k. In this paper, the existence of a preferred basis for a singular M_∨- matrix A = sI − B with index(B) ≤ 1 is proven. Some equivalent conditions for the equality of the height and level characteristics of A are studied. The well structured property of the reduced graph of A is discussed. Also the possibility of the existence of preferred basis for another generalization of M-matrices, known as GM-matrices, is studied.
Saha, Manideepa and Bandopadhyay, Sriparna.
"Combinatorial properties of generalized M-matrices",
Electronic Journal of Linear Algebra,
Volume 30, pp. 550-576.