Nonconcavity of the Spectral Radius in Levinger's Theorem

TL;DR

This paper investigates the concavity of the spectral radius function in Levinger's theorem, providing counterexamples to general concavity and identifying specific matrix classes where it holds.

Contribution

It disproves the general concavity claim for spectral radius in Levinger's theorem and establishes conditions under which concavity is valid for certain matrix classes.

Findings

Counterexamples show spectral radius is not always concave.

Concavity holds for 2x2, weighted shift, and certain Toeplitz matrices.

Open problem: characterizing matrices with concave spectral radius in Levinger's homotopy.

Abstract

Let ${\bf A} \in R^{n \times n}$ be a nonnegative irreducible square matrix and let $r({\bf A})$ be its spectral radius and Perron-Frobenius eigenvalue. Levinger asserted and several have proven that $r(t):=r((1{-}t) {\bf A} + t {\bf A}^\top)$ increases over $t \in [0,1/2]$ and decreases over $t \in [1/2,1]$. It has further been stated that $r(t)$ is concave over $t \in (0,1)$. Here we show that the latter claim is false in general through a number of counterexamples, but prove it is true for ${\bf A} \in R^{2\times 2}$, weighted shift matrices (but not cyclic weighted shift matrices), tridiagonal Toeplitz matrices, and the 3-parameter Toeplitz matrices from Fiedler, but not Toeplitz matrices in general. A general characterization of the range of $t$, or the class of matrices, for which the spectral radius is concave in Levinger's homotopy remains an open problem.