LAUSR

Abstract Let A ∈ R n × 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 + t A ⊤ ) 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 ∈ R 2 × 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.