Nonconcavity of the spectral radius in Levinger's theorem

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Let A∈Rn×n be a nonnegative irreducible square matrix and let r(A) be its spectral radius and Perron-Frobenius eigenvalue. Levinger asserted and several have proven that r(t):=r((1−t)A+tA⊤) increases over t∈[0,1/2] and decreases over t∈[1/2,1]. It has further been stated that r(t) is concave over t∈(0,1). Here we show that the latter claim is false in general through a number of counterexamples, but prove it is true for A∈R2×2, 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.