论文标题
Levinger定理中频谱半径的非conemavity
Nonconcavity of the Spectral Radius in Levinger's Theorem
论文作者
论文摘要
令$ {\ bf a} \ in r^{n \ times n} $为不可不可约的方形矩阵，让$ r（{\ bf a}）$为其频谱半径和perron-frobenius eigenvalue。 Levinger断言，有几个事实证明，$ r（t）：= r（（（1 { - } t）{\ bf a} + t {\ bf a}^\ top）$增加了[0,1/2] $的$ t \ in [0,1/2] $，并且在[1/2,1] $中$ t \ in [0,1/2] $。进一步指出，$ r（t）$超过$ 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矩阵一般。 $ t $或矩阵类别的范围的一般表征，光谱半径在Levinger的同型中是凹面的，仍然是一个空旷的问题。
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.
