On the Toeplitz embedding of an arbitrary matrix
type: Item
instance of: scholarly article
title: On the Toeplitz embedding of an arbitrary matrix
zbMATH Open document ID: 0538.15008
DOI: 10.1016/0024-3795(83)90151-9
author: Y. Genin
author: Y. Kamp
author: Philippe Delsarte
published in: Linear Algebra and its Applications
publication date: 1983-01-01T00:00:00Z
review text: In the first part of the paper the authors give a thorough analysis of the various polynomials associated with the inverse of the upper left principal minors of a block Toeplitz matrix. This provides a unified framework for deriving the basic formulae of the Levinson and the Trench algorithms for inverting block Toeplitz matrices; in case of ordinary Toeplitz matrices these algorithms require \(O(n^ 2)\) arithmetic operations (see e.g. the paper of \textit{B. Friedlander}, \textit{M. Morf}, \textit{I. Kailath} and \textit{L. Ljung} [ibid. 27, 31-60 (1979; Zbl 0414.15005)]). Another important consequence of the polynomial approach is the generalization of an earlier result of the authors on the Schur parametrization of positive definite block Toeplitz matrices [see SIAM J. Appl. Math. 36, 34-46 (1979; Zbl 0417.42013)]. In the second part the authors use the method of Delosme and Morf for the embedding of an arbitrary block matrix into a block Toeplitz matrix, and show that all results of the previous part can be carried over to arbitrary block matrices.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 65F05
zbMATH DE Number: 3855299
zbMATH Keywords: Toeplitz embedding
zbMATH Keywords: block Toeplitz matrices
zbMATH Keywords: Schur parameters
zbMATH Keywords: Christoffel-Darboux formula
zbMATH Keywords: Levinson algorithm
zbMATH Keywords: Trench algorithms
MaRDI profile type: Publication
full work available at URL: 0024-3795(83)90151-9
OpenAlex ID: W2037739163
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description: scientific article; zbMATH DE number 3855299