More on Loewner matrices
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.
generalized rational interpolation problem
scientific article; zbMATH DE number 614154
scientific article