Spectral properties of sums of certain Kronecker products
type: Item
instance of: scholarly article
title: Spectral properties of sums of certain Kronecker products
zbMATH Open document ID: 1191.15007
author: Yimin Wei
author: Jun-e Feng
author: James Lam
published in: Linear Algebra and its Applications
publication date: 2009-09-14T00:00:00Z
review text: The authors study spectral properties of matrices of the form \(A\otimes A + C\otimes C\) and \(I\otimes A + A\otimes I + C\otimes C\), where \(A\) and \(C\) are real square matrices, and \(I\) the identity matrix. These matrices are related to the linear endomorphisms \({\mathcal T}_{A,C}:Z\mapsto AZA^T+CZC^T\) respectively \(\widetilde{\mathcal T}_{A,C}:Z\mapsto AZ+ZA^T+CZC^T\) of the matrix algebra \(\mathbb{C}^{n\times n}\). The main results are Theorem~3 and Theorem~5. Theorem~3 asserts that the spectral radius of \({\mathcal T}_{A,C}\) equals the spectral radius of its restriction to the space \(\mathbb{S}^{n\times n}\subset\mathbb{C}^{n\times n}\) of symmetric matrices. Similarly, by Theorem~5, the spectral abscissa (\(\max\{\mathop{\mathrm{Re}}\lambda \mid \lambda\in\mathbb{C} \text{ is an eigenvalue}\}\)) of \(\widetilde{\mathcal T}_{A,C}\) is the same as the spectral abscissa of the restriction \(\widetilde{\mathcal T}_{A,C}|_{\mathbb{S}^{n\times n}}\). The presentation is rather fuzzy. Notably, Theorem~2 is really just a complicated statement (followed by an equally complicated proof) of the fact that the respective spaces of symmetric and anti-symmetric matrices are invariant under \({\mathcal T}_{A,C}\).
reviewed by: Erik Darpö
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A69
zbMATH DE Number: 5603586
zbMATH Keywords: tensor product
zbMATH Keywords: spectral radius
zbMATH Keywords: spectral abscissa
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenvector
zbMATH Keywords: Kronecker product
MaRDI profile type: Publication
full work available at URL: j.laa.2009.06.004
OpenAlex ID: W2083756075
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DOI: 10.1016/J.LAA.2009.06.004
description: scientific article; zbMATH DE number 5603586