On the number of matrices to generate a matrix \(\ast\)-algebra over the real field
type: Item
instance of: scholarly article
title: On the number of matrices to generate a matrix \(\ast\)-algebra over the real field
zbMATH Open document ID: 1277.15012
author: Daishi Aiura
author: Naonori Kakimura
author: Kazuo Murota
published in: Linear Algebra and its Applications
publication date: 2013-01-16T00:00:00Z
review text: Let \(M_n(\mathbb{R})\) be the set of matrices of order \(n\) over the real field \(\mathbb{R}\). A subset \(\mathcal{T}\) of \(M_n(\mathbb{R})\) is said to be a matrix \(*\)-algebra if \(I_n \in \mathcal{T}\) and \(\mathcal{T}\) is closed under linear combination, multiplication, and transposition. In the present paper, among other things, it is shown that any matrix \(*\)-algebra over \(\mathbb{R}\) can be generated with at most four randomly-chosen symmetric matrices. The proof of this characterization result involves a skillful use of a structure theorem for matrix \(*\)-algebras and the notion of genericity in the eigenvalue structure.
reviewed by: Jaydeb Sarkar
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 6125863
zbMATH Keywords: matrix \(\ast\)-algebra
zbMATH Keywords: minimal generators
zbMATH Keywords: structure theorem
zbMATH Keywords: symmetric matrices
zbMATH Keywords: eigenvalue structure
MaRDI profile type: Publication
full work available at URL: j.laa.2012.08.022
OpenAlex ID: W2011073130
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DOI: 10.1016/J.LAA.2012.08.022
description: scientific article; zbMATH DE number 6125863