On the construction of a Jacobi matrix from its mixed-type eigenpairs
type: Item
instance of: scholarly article
title: On the construction of a Jacobi matrix from its mixed-type eigenpairs
zbMATH Open document ID: 1025.65024
DOI: 10.1016/S0024-3795(02)00488-3
author: Zhen-Yun Peng
author: Xiyan Hu
author: Lei Zhang
published in: Linear Algebra and its Applications
publication date: 2003-03-11T00:00:00Z
review text: The problem considered by the authors is to recover a Jacobi matrix \(J\) of finite size, \(n\times n,\) from a given set of eigenpairs associated with two principal submatrices. Observe that the Jacobi matrix \(J\) requires \(2n-1\) entries while the data used for the recovery is given by \(\lambda, \mu_1, \mu_2 \) and \[ \begin{aligned} {\mathbf Y}_1 &=(y_{1},y_{2},\dots,y_{k}) \\ {\mathbf {Y}_{2}} &=(y_{k+1},y_{k+2},\dots,y_{n}) \\ {\mathbf X} &=( x_{1},x_{2},\dots,x_{n})\end{aligned} \] where \(J{\mathbf X=}\lambda {\mathbf X}\), \(J_{i}{\mathbf {Y}_{i}}=\mu _{i} {\mathbf {Y}_{i}}\quad i=1,2\) and \(J_{i}\) are submatrices. After rescaling the vectors \({\mathbf Y}_{i}\), \({\mathbf X}\), the data reduces to \(2n\). The authors translate the inverse problem into a sequential algebraic system. They also manage, by using \(2\times 2\) determinants only, to find explicit conditions for the existence of a unique solution. The proof yields an algorithm for the reconstruction of the Jacobi matrix and numerical examples are provided in the last section.
Mathematics Subject Classification ID: 65F18
Mathematics Subject Classification ID: 65H17
zbMATH DE Number: 1880074
zbMATH Keywords: Jacobi matrix
zbMATH Keywords: inverse eigenvalue problem
zbMATH Keywords: algorithm
zbMATH Keywords: reconstruction
zbMATH Keywords: numerical examples
reviewed by: Amin Boumenir
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880074