Nonconcavity of the Spectral Radius in Levinger's Theorem

Abstract

Let $\mathbf{A}\in R^{n\times n}$ be a nonnegative irreducible square matrix and let $r(\mathbf{A})$ be its spectral radius and Perron-Frobenius eigenvalue. Levinger asserted and several have proven that $r(t):=r((1-t)\mathbf{A}+t{\mathbf{A}}^{\top })$ 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 $\mathbf{A}\in R^{2\times 2}$, 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.

v4: Dedication and biographical note. v3: Includes reviewer suggestions. Accepted to Linear Algebra and Its Applications. v2: Replaced graphics that had buggy PDF. 19 pages, 6 figures

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