On the Asymptotic Behavior of the Perron Eigenvalue

Abstract

A real n×n matrix A = (a_ij) is said to be an ml-matrix if a_ij ≥ 0 for all i ≠ j. Such matrices arise in the study of stability of multiple markets in mathematical economics [1], [3] as well as in the theory of finite Markov processes [5]. As can be seen through the Perron-Frobenius theorem, any ml-matrix has a real eigenvalue which is called the Perron eigenvalue of A, so that if λ is any other eigenvalue of A then . Applications for this eigenvalue can be found in the above areas.

In this paper, the asymptotic behavior of the Perron eigenvalue is studied. In particular, if A is an n×n ml-matrix and an n× nonnegative matrix, bounds are found for . Further, the asymptotic behavior of approaches infinity, is studied.