一类对称双线性型下三对角矩阵特征值的谱分隔性质

The spectrum separation theory of a kind of tridiagonal matrix in the symmetric bilinear form

本文给出了一类具有互异实特征值的三对角矩阵,并在对称双线性型下定义了其对称性,这种对称性称之为伪对称。当给其一个秩1的扰动时,分析了扰动前后的两个矩阵的特征多项式之间的关系。通过借助具有互异的实特征值的伪对称三对角矩阵可对角化且其特征向量的第一个和最后一个分量均不为零这一理论,讨论了扰动前后的矩阵的谱分隔性质。

This paper presents a class of tridiagonal matrices with distinct eigenvalues and defines their symmetry under the symmetric bilinear form,referred to as pseudo-symmetry.The relationship between the characteristic polynomials of the matrices before and after a rank-1 perturbation is analyzed.By utilizing the theory of diagonalization for pseudo-symmetric tridiagonal matrices with distinct real eigenvalues and non-zero first and last components of their eigenvectors,the spectral separability properties of the matrices before and after the perturbation are discussed.