More on Loewner matrices
type: Item
instance of: scholarly article
title: More on Loewner matrices
zbMATH Open document ID: 0814.15012
DOI: 10.1016/0024-3795(94)90206-2
author: H. X. Zhang
author: Gongning Chen
published in: Linear Algebra and its Applications
publication date: 1994-08-15T00:00:00Z
review text: Generalized Loewner matrices arise in the study of the generalized rational interpolation problem. The present paper is divided into five sections. Section 1 deals with basic definitions and preliminary results. In Section 2 the authors derive a connection between generalized Loewner matrices and Bézoutians, originally established by \textit{A. C. Anderson} and \textit{B. D. O. Antoulas} [IMA J. Math. Control Inf. 3, 61-88 (1986; Zbl 0637.93014)]. Using this connection they establish some properties of Loewner matrices, in particular Barnett-like factorization formulas and the intertwining characterization of Loewner matrices. In Section 3 they study relations between Loewner and Hankel matrices and, in particular, they give a new simple derivation of a result of \textit{Z. Vavrin} [Linear Algebra Appl. 143, 171-222 (1991; Zbl 0712.15023)]. In Section 4 they introduce a class of matrices \((R\)-matrices) which can be written in the form \(R = BHA^ t\), where \(H\) is a Hankel matrix and \(A\) and \(B\) are invertible matrices which depend on a pair of interpolation vectors and a polynomial compatible with \(H\). A Barnett-like formula and an intertwining relation for \(R\)-matrices are studied. The authors also prove that the inverse of a nonsingular \(R\)-matrix is a Loewner matrix. The final section deals with products of Loewner and \(R\)-matrices.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 614154
zbMATH Keywords: generalized Loewner matrices
zbMATH Keywords: \(R\)-matrices
zbMATH Keywords: generalized rational interpolation problem
zbMATH Keywords: Bézoutians
zbMATH Keywords: factorization
zbMATH Keywords: Hankel matrices
reviewed by: Fergus J. Gaines
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90206-2
OpenAlex ID: W2058919354
cites work: On the Scalar Rational Interpolation Problem
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cites work: Inverses of Löwner matrices
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cites work: A unified approach to Loewner and Hankel matrices
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description: scientific article; zbMATH DE number 614154