论文标题
特里迪亚尔（Tridiagonal）遗传性toeplitz矩阵的特征值和偏角角扰动
Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners
论文作者
论文摘要
在本文中，我们研究了第一列中的遗体toeplitz矩阵的特征值$ 2，-1,0，\ ldots，0，-α$。请注意，生成符号取决于矩阵的$ n $。如果$ |α| \ le 1 $，则特征值属于$ [0,4] $，并且在$ [0，π] $上的函数$ g（x）= 4 \ sin^2（x/2）$。当$ |α|> 1 $和$ n $倾向于无限时，情况发生了巨大变化。然后，两个极端特征值（最小和最大值）均由$ [0,4] $铺设，并迅速收敛于由$α$的值确定的某些限制，而其他所有其他属性则属于$ [0,4] $，并且非属于$ g $。在所有情况下，我们都会将特征方程式转换为一种方便地通过数值方法求解的形式，并为特征值推导渐近公式。
In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-α$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|α|\le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,π]$. The situation changes drastically when $|α|>1$ and $n$ tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of $[0,4]$ and converge rapidly to certain limits determined by the value of $α$, whilst all others belong to $[0,4]$ and are asymptotically distributed as $g$. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues.
