Geodesic
On extremal properties of the dominant eigenvalue
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1019-1034.

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The property of almost monotonicity for the non-singular irreducible M-matrix is specified. In its existing form the property means that the result of application of the above matrix to a vector is either the zero vector or a vector with at least one component positive and one component negative. In this paper the positive and the negative components are explicitly indicated. As an application, a criterion of Pareto-extremality for a vector function with essentially non-negative matrix of partial derivatives is derived. The criterion is a counterpart of the classical Fermat theorem on vanishing of the derivative in an extremal point of a function. The proofs are based on geometric properties of $n$-dimensional simplex described in two lemmas of independent nature. 
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L. I. Krechetov. On extremal properties of the dominant eigenvalue. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1019-1034. http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a5/

[1] Berman A., Plemmons R. J., Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979 | MR | Zbl

[2] Seneta E., Non-negative matrices, Wiley, New York, 1973 | MR

[3] Khorn R., Dzhonson Ch., Matrichnyi analiz, Mir, M., 1989 | MR