Random evolutions and the spectral radius of a non - negative matrix

1. Introduction and summary. This paper offers yet another example of what probability theory can do for analysis. Using a Feynman-Kac formula derived in the theory of random evolutions (51, we find an expression (1) for the spectral radius r(A) of a finite square non-negative matrix A. This expression makes it very easy to study how r(A) behaves as a function of the diagonal elements of A. Kac (7) derived an expression of the same form as (1) for the principal eigenvalue of a second-order ordinary differential equation, using a Feynman-Kac formula for Brownian motion rather than for a finite-state Markov chain. His result has been extensively generalized (Donsker and Varadhan (3)). A direct derivation of (1) for non-negative matrices and the two main consequences of (1) derived in section 2 (inequalities (7) and (9)) may be new. Inequality (7) is a lower bound for r(A) when A is irreducible. Inequality (9) asserts that, whether A is irre-ducible or not, r(A) is a convex function of the main diagonal of A. Section 3 reviews alternative, partially successful approaches to the same results. This paper substantially generalizes the major results of (2) and provides much