Factorizing matrices by Dirichlet multiplication
type: Item
instance of: scholarly article
title: Factorizing matrices by Dirichlet multiplication
zbMATH Open document ID: 1263.15013
author: Artur Sowa
published in: Linear Algebra and its Applications
publication date: 2013-02-22T00:00:00Z
review text: A Dirichlet multiplier is a matrix \[ \left [\begin{matrix} a_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots \\ a_2 & a_1 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots \\ a_3 & 0 & a_1 & 0 & 0 & 0 & 0 & 0 & \ldots \\ a_4 & a_2 & 0 & a_1 & 0 & 0 & 0 & 0 & \ldots \\ a_5 & 0 & 0 & 0 & a_1 & 0 & 0 & 0 & \ldots \\ a_6 & a_3 & a_2 & 0 & 0 & a_1 & 0 & 0 & \ldots \\ a_7 & 0 & 0 & 0 & 0 & 0 & a_1 & 0 & \ldots \\ a_8 & a_4 & 0 & a_2 & 0 & 0 & 0 & a_1 & \ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \right] . \] where \(a_1, a_2, \ldots\) is a sequence of numbers. Using finite submatrices of Dirichlet multipliers, the author shows that any nonsingular matrix is a product of Dirichlet multipliers. He also gives an efficient algorithm for a numerical factorization of such a matrix.
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B99
Mathematics Subject Classification ID: 65F50
zbMATH DE Number: 6138422
zbMATH Keywords: Dirichlet multiplication
zbMATH Keywords: matrix factorization
zbMATH Keywords: Dirichlet series
zbMATH Keywords: fast transform
zbMATH Keywords: lifting schemas
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: j.laa.2012.09.021
OpenAlex ID: W1968522278
cites work: Q4101884
cites work: The collected works of Arne Beurling. Volume 1: Complex analysis. Volume 2: Harmonic analysis. Ed. by Lennart Carleson, Paul Malliavin, John Neuberger, John Wermer
cites work: Q5621389
cites work: A Hilbert space of Dirichlet series and systems of dilated functions in \(L^ 2(0,1)\)
cites work: Numerical recipes in C++. The art of scientific computing
cites work: On an eigenvalue problem with a reciprocal-linear term
cites work: The Dirichlet ring and unconditional bases in \(L_2[0, 2\pi]\)
DOI: 10.1016/J.LAA.2012.09.021
description: scientific article; zbMATH DE number 6138422