FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 4, PAGES 1019-1034

**On extremal properties of the dominant eigenvalue**

L. I. Krechetov

Abstract

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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.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k02/k024/k02406h.htm

Last modified: April 10, 2003