Circulants, displacements and decompositions of matrices
type: Item
instance of: scholarly article
title: Circulants, displacements and decompositions of matrices
zbMATH Open document ID: 0772.15010
DOI: 10.1007/BF01200697
published in: Integral Equations and Operator Theory
publication date: 1993-06-29T00:00:00Z
review text: Following \textit{T. Kailath}, \textit{S.-Y. Kung} and \textit{M. Morf} [J. Math. Anal. Appl. 68 395-407 (1979; Zbl 0433.15001)] the authors introduce in \(\mathbb{C}^{n\times n}\) the linear operator, which transforms each matrix \(A\) in its displacement \(\nabla_ 0(A)=A-Z_ 0AZ^ T_ 0\), where \[ Z_ 0=\left[\begin{matrix} 0 &\ldots &\ldots &\ldots &0 \\ 1 &0 & & &\vdots \\ 0 & 1 &\ddots & &\vdots \\ \vdots &\ddots &\ddots & \ddots & \vdots \\ 0 &\ldots & 0 & 1 & 0 \end{matrix} \right] \] is the lower shift matrix. New formulas for the representation of matrices and their inverses in the form of sums of products of factor circulants are suggested, which are based on the analysis of the factor cyclic displacement of matrices. The results in applications to Toeplitz matrices generalize the formulas of \textit{I. C. Gohberg} and \textit{A. A. Semencul} [Mat. Issled. 7, No. 2(24), 201-223 (1972; Zbl 0288.15004)], \textit{A. Ben-Artzi} and \textit{T. Shalom} [Linear Algebra Appl. 75, 173-192 (1986; Zbl 0586.15005)], and \textit{G. Heinig} and \textit{K. Rost} [Algebraic methods for Toeplitz-like matrices and operators (Akademie-Verlag, Berlin, 1984; Zbl 0549.15013)] and are useful for complexity analysis.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 68Q25
zbMATH DE Number: 222146
zbMATH Keywords: decompositions
zbMATH Keywords: inversion of matrices
zbMATH Keywords: displacement
zbMATH Keywords: factor circulants
zbMATH Keywords: factor cyclic displacement
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: complexity analysis
author: I. Gohberg
author: V. Olshevsky
reviewed by: Yueh-er Kuo
MaRDI profile type: Publication
cites work: Q5751848
cites work: On inversion of Toeplitz and close to Toeplitz matrices
cites work: New inversion formulas for matrices classified in terms of their distance from Toeplitz matrices
cites work: Displacement operator based decompositions of matrices using circulants or other group matrices
cites work: Q4775961
cites work: Algebraic methods for Toeplitz-like matrices and operators
cites work: Displacement ranks of matrices and linear equations
full work available at URL: https://doi.org/10.1007/bf01200697
OpenAlex ID: W2047954521
description: scientific article; zbMATH DE number 222146