On extremal properties of the dominant eigenvalue

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.