Direct approach to the band completion problem
type: Item
instance of: scholarly article
title: Direct approach to the band completion problem
zbMATH Open document ID: 1054.15018
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: The paper deals with the partial band matrix completion problem, that is, given a partial band matrix \(A\), is there a completion \(A_c\) of \(A\) whose inverse is a band matrix? The problem considered in this paper concerns the completion of a partial block matrix with a specified band. The completion is required to be invertible and the inverse to be a band matrix of the same width as the specified band of the original matrix. The completion given by the authors is made directly via explicit formulas. They show that the solution of the band completion problem coincides with the problem of completion of a band to a Green matrix. This result is based on a theorem on relations between the ranks of submatrices of a block matrix and the ranks of the corresponding submatrices of its inverse. In addition, the completion obtained belongs to the class of quasiseparable matrices. At the end of the paper the authors study two special examples. The first is concerned with the Toeplitz case and the second contains a block tridiagonal case and the problem of completion of a special matrix which is connected with the problem of completions of triangular parts of matrices to contractions.
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 2083514
zbMATH Keywords: band completion problem
zbMATH Keywords: Green matrices
zbMATH Keywords: banded inverses
zbMATH Keywords: structured matrices
zbMATH Keywords: block tridiagonal matrix
zbMATH Keywords: partial block matrix
zbMATH Keywords: band matrix
zbMATH Keywords: Toeplitz matrix
author: Y. Eidelman
author: I. Gohberg
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2003.07.014
OpenAlex ID: W2036508582
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DOI: 10.1016/J.LAA.2003.07.014
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description: scientific article; zbMATH DE number 2083514