Matrix inversion cases with size-independent tensor rank estimates
type: Item
instance of: scholarly article
title: Matrix inversion cases with size-independent tensor rank estimates
zbMATH Open document ID: 1180.15005
published in: Linear Algebra and its Applications
publication date: 2009-07-02T00:00:00Z
review text: Over an arbitrary field let \(M\) be a nonsingular \(pq\times pq\) matrix of the form \[ M=I+A \otimes X+Y \otimes B, \] where \(A,Y\) are \(p\times p\) and \(X,B\) are \(q\times q\). Assume additionally that rank \(X = \text{rank } Y = 1\) and that \(I+A\otimes X\) and \(I+Y\otimes B\) are non-singular. It is shown that \(M^{-1}\) can be written in the form \[ M^{-1}=I +\hat A\otimes X +Y\otimes \hat B + Y_1\otimes X_1 + Y_2\otimes X_2 \] for some matrices \(\hat A, \hat B\) and matrices \(X_1,X_2,Y_1,Y_2\) of rank at most \(1\). This implies that the tensor rank of \(M^{-1}\) is bounded from above by 5 independently of \(p\) and \(q\). For matrices over the complex numbers a stronger version of the result is established.
reviewed by: Hans Havlicek
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 5574547
zbMATH Keywords: tensor rank
zbMATH Keywords: Kronecker product
zbMATH Keywords: low-rank matrices
zbMATH Keywords: inverse matrices
zbMATH Keywords: multilevel matrices
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: circulant matrices
author: Ivan Oseledets
author: E. E. Tyrtyshnikov
author: N. L. Zamarashkin
MaRDI profile type: Publication
full work available at URL: j.laa.2009.03.001
OpenAlex ID: W2093692490
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DOI: 10.1016/J.LAA.2009.03.001
description: scientific article; zbMATH DE number 5574547