Nonnegative $\lambda $-Monotone Matrices
S. K. Jain, Larry E. Snyder · SIAM Journal on Algebraic and Discrete Methods · 1981
In this paper, the structure of nonnegative $m \times n$ matrices A satisfying $AXA = A$, for some nonnegative $n \times m$ matrix X, is obtained. Several equivalent characterizations of such matrices A have been given earlier by Plemmons [Proc. Amer. Math. Soc., 39 (1973), pp. 26–32] and Berman–Plemmons [Linear and Multlinear Algebra, 2 (1974), pp. 161–172]. The structure of matrices given in this paper unifies all the previous known results on $\lambda $-monotone matrices where $1 \in \lambda $. The importance of $\lambda $-monotonicity to problems in mathematical economics, in probability and statistics, and in numerical linear algebra, is documented in a recent book by Berman and Plemmons [Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979].