Convexity and log convexity for the spectral radius

The starting point of this paper is a theorem by J. F. C. Kingman which asserts that if the entries of a nonnegative matrix are log convex functions of a variable then so is the spectral radius of the matrix. A related result of J. Cohen asserts that the spectral radius of a nonnegative matrix is a convex function of the diagonal elements. The first section of this paper gives a new, unified proof of these results and also analyzes exactly when one has strict convexity. The second section gives some very simple proofs of results of Friedland and Karlin concerning “min-max” characterizations of the spectral radius of nonnegative matrices. These arguments also yield, as will be shown in another paper, min-max characterizations of the principal eigenvalue of second order elliptic boundary value problems on bounded domains. The third section considers the cone K of nonnegative vectors in Rⁿ and continuous maps f: K → K which are homogeneous of degree one and preserve the partial order induced by K. The (cone) spectral radius of such maps is defined and a direct generalization of Kingman's theorem to a subclass of such nonlinear maps is given. The final section of this paper treats a problem that arises in population biology. If K₀ denotes the interior of K and f is as above, when can one say that f has a unique eigenvector (to within normalization) in K₀? A subtle point to be noted is that f may have other eigenvectors in the boundary of K. If u ϵ K₀ is an eigenvector of f, |u| = 1, and g(x) = f(x)/|f(x)|, when can one say that for any x ϵ K₀, g^p(x), the pth iterate of g acting on x, converges geometrically to u? The fourth section provides answers to these questions that are adequate for many of the population biology problems.