On the additive structure of the inverses of banded matrices
type: Item
instance of: scholarly article
title: On the additive structure of the inverses of banded matrices
zbMATH Open document ID: 0598.15004
DOI: 10.1016/0024-3795(86)90281-8
published in: Linear Algebra and its Applications
publication date: 1986-01-01T00:00:00Z
review text: Using the results of \textit{W. W. Barrett} and \textit{P. J. Feinsilver} [ibid. 41, 111-130 (1981; Zbl 0472.15004)] the author proves the following main result: Let B be an \(n\times n\) proper \((2k+1)\)-diagonal matrix. Then \(B^{-1}=\sum^{k}_{p=1}[(u_ pv^ T_ p)\circ E_ n+(x_ py^ T_ p)\circ L_ n],\) where \(u_ p,v_ p,x_ p,y_ p\) are n-vectors, \(E_ n\underline\triangle (e_{ij})\), \(e_{ij}\underline\triangle 1\) if \(i\leq j\), and \(\underline\triangle 0\) if \(i>j\), \(L_ n\underline\triangle E^ T_ n-I_ n\) \((I_ n\) is the unit matrix) and \(\circ\) denotes the Hadamard product. If in addition B is symmetric, then \(B^{-1}\) may be expressed as a sum of k symmetric matrices belonging to the class of inverses of symmetric irreducible tridiagonal matrices. Two examples are given. The results may be applied for the resolution of constant coefficient banded linear systems in VLSI models.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A06
zbMATH DE Number: 3964052
zbMATH Keywords: band matrices
zbMATH Keywords: inverses
zbMATH Keywords: symmetric irreducible tridiagonal matrices
zbMATH Keywords: banded linear systems
author: Francesco Romani
reviewed by: Mihail Voicu
MaRDI profile type: Publication
full work available at URL: 0024-3795(86)90281-8
OpenAlex ID: W2054467064
cites work: Inverses of Matrices $\{a_{ij}\}$ which Satisfy $a_{ij} = 0$ for $j &gt; i+p$.
cites work: A theorem on inverses of tridiagonal matrices
cites work: Inverses of banded matrices
cites work: Q4120046
cites work: Methods of inverting tridiagonal matrices
cites work: On some properties of a tridiagonal and penta-diagonal matrix
cites work: VLSI implementation of fast solvers for band linear systems with constant coefficient matrix
cites work: Q5543004
cites work: Brève communication. Inversion de matrices tridiagonales symétriques
cites work: Inversion of band matrices
description: scientific article; zbMATH DE number 3964052