On an inverse eigenvalue problem for unitary Hessenberg matrices
type: Item
instance of: scholarly article
title: On an inverse eigenvalue problem for unitary Hessenberg matrices
zbMATH Open document ID: 0826.15008
DOI: 10.1016/0024-3795(93)00188-6
author: G. S. Ammar
author: Chunyang He
published in: Linear Algebra and its Applications
publication date: 1995-12-03T00:00:00Z
review text: The authors consider the inverse eigenvalue problem for unitary Hessenberg matrices. It is proved that if \(\{\lambda_k\}^n_{k = 1}\) and \(\{\mu_k\}^{n - 1}_{k = 0}\) are two sets of strictly interlacing points on the unit circle in the complex plane, then there exists a unique unitary Hessenberg matrix \(H = H (\gamma_1, \gamma_2, \ldots, \gamma_n)\) such that the spectrum of the matrix \(\lambda (H) = \{\lambda_k\}^n_{k = 1}\) and \(\lambda (H_{n - 1}') = \{\mu_k\}^{n - 1}_{k = 1}\), where \(H_{n - 1}' = H (\gamma_1, \gamma_2, \ldots, \gamma_{n - 2}, \rho_{n - 1})\) and \(\rho_{n - 1} = (\gamma_{n - 1} + \overline {\mu}_0 \gamma_n)/(1 + \overline {\mu}_0 \overline {\gamma}_{n - 1} \gamma_n)\). A procedure for constructing the matrix \(H (\gamma_1, \gamma_2, \ldots, \gamma_n)\) is presented.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 753874
zbMATH Keywords: principal submatrix
zbMATH Keywords: Schur parameter
zbMATH Keywords: inverse eigenvalue problem
zbMATH Keywords: unitary Hessenberg matrices
reviewed by: Cheng-Xian Xu
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)00188-6
OpenAlex ID: W1985109612
cites work: Q3974761
cites work: A survey of matrix inverse eigenvalue problems
cites work: On a Sturm Sequence of Polynomials for Unitary Hessenberg Matrices
cites work: Tridiagonal Approach to the Algebraic Environment of Toeplitz Matrices, Part I: Basic Results
cites work: Tridiagonal Approach to the Algebraic Environment of Toeplitz Matrices, Part II: Zero and Eigenvalue Problems
cites work: Q3359644
cites work: Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle
cites work: The numerically stable reconstruction of Jacobi matrices from spectral data
cites work: A divide and conquer method for unitary and orthogonal eigenproblems
cites work: Discrete Least Squares Approximation by Trigonometric Polynomials
description: scientific article; zbMATH DE number 753874