A divide-and-conquer method for constructing a pseudo-Jacobi matrix from mixed given data
A divide-and-conquer method for constructing a pseudo-Jacobi matrix from mixed given data
复制标题
DOI：
10.1016/j.laa.2023.05.026
复制
发表时间：
2023-06
影响因子：
1.1
通讯作者：
Wei-Ru Xu;N. Bebiano;Q. Shu;Ting-ting Feng
中科院分区：
文献类型：
--
作者：
Wei-Ru Xu;N. Bebiano;Q. Shu;Ting-ting Feng
对于给定的签名算子H= I r − I n− r，伪雅可比矩阵是相对于对称双线性形式<H，H>的自伴矩阵，并且它是经典雅可比矩阵在不定标积空间设置的对应。在这篇文章中，我们考虑这类矩阵的逆特征值问题。主要问题是从规定的不同真实的数的n元组和阶不小于n 2的Jacobi矩阵构造n× n伪Jacobi矩阵，使得其谱是该元组，并且给定的Jacobi矩阵是其尾随主子矩阵。采用分治策略求解该问题，并给出了问题可解的充分必要条件。根据所得结果设计了求解该伪Jacobi矩阵特征值反问题的数值算法。文中还给出了一些数值例子来验证重构算法的正确性。
For the given signature operator H= I r⊕− I n− r, a pseudo-Jacobi matrix is a self-adjoint matrix relatively to a symmetric bilinear form<⋅,⋅> H, and it is the counterpart of a classical Jacobi matrix to the indefinite scalar product space setting. In this article, we consider an inverse eigenvalue problem for this class of matrices. The main concern is to construct an n× n pseudo-Jacobi matrix from a prescribed n-tuple of distinct real numbers and a Jacobi matrix of order not less than⌊ n 2⌋, such that its spectrum is this tuple and the given Jacobi matrix is its trailing principal submatrix. A divide-and-conquer scheme is used to solve this problem, and a necessary and sufficient condition under which the problem is solvable is presented. A numerical algorithm is designed to solve this pseudo-Jacobi matrix inverse eigenvalue problem according to the obtained results. Some illustrative numerical examples are also given to test the reconstructive algorithm.