Multiple LU factorizations of a singular matrix
type: Item
instance of: scholarly article
title: Multiple LU factorizations of a singular matrix
zbMATH Open document ID: 1106.15009
author: Froilán M. Dopico
author: Charles R. Johnson
author: Juan M. Molera
published in: Linear Algebra and its Applications
publication date: 2006-12-07T00:00:00Z
review text: A general \(m\)-by-\(n\) matrix \(A=(A_{i,j})\) over the complex field \(\mathbb C\) has an LU factorization if it can be written as \(A=LU\) with \(L\) \(m\)-by-\(m\) lower triangular (\(L_{ij}=0\) whenever \(i < j\)) and \(U\) \(m\)-by-\(n\) upper triangular (\(U_{ij}=0\) whenever \(i > j\)). A nonsingular \(L\) can be normalized to have 1's on the diagonal and is called unit lower triangular. The authors describe all possible LU factorizations, with \(L\) nonsingular unit lower triangular, of any singular matrix \(A\) for which such a factorization exists. Any matrix \(A\) is factorized under left multiplication by unit lower triangular matrices. In this way a new canonical form, whose entries can be determined in terms of quotients of minors of \(A\), is introduced. This canonical form allows easily to characterize the matrices having an LU factorization. Formulae, in terms of quotients of minors of \(A\), the denominators of which are nonzero, are given for those entries of \(L\) and \(U\) that are uniquely determined.
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 65F05
zbMATH DE Number: 5077724
zbMATH Keywords: canonical form
zbMATH Keywords: LU factorization
zbMATH Keywords: minors
zbMATH Keywords: singular matrices
describes a project that uses: Matlab
describes a project that uses: mctoolbox
MaRDI profile type: Publication
full work available at URL: j.laa.2006.03.043
OpenAlex ID: W2166815216
Wikidata QID: Q114152450
cites work: Q4223568
cites work: Accuracy and Stability of Numerical Algorithms
cites work: Matrix Analysis
cites work: Q4344146
cites work: $LU$ factorizations
DOI: 10.1016/J.LAA.2006.03.043
description: scientific article; zbMATH DE number 5077724