Spaces of skew-symmetric matrices satisfying \(A^3=\lambda A\)
type: Item
instance of: scholarly article
title: Spaces of skew-symmetric matrices satisfying \(A^3=\lambda A\)
zbMATH Open document ID: 1261.15023
author: Yunus Özdem{i}r
published in: Linear Algebra and its Applications
publication date: 2013-01-16T00:00:00Z
review text: Let \(m\in \mathbb N\). We may write \(m=(2a+1)\cdot 2^{4b+c}\) where \(a,b\) and \(c\) are non-negative integers and \(c<4\). The Radon-Hurwitz number \(\rho (m)\) of \(m\) is defined as \(\rho (m)=2^{c}+8b\). The Radon-Hurwitz numbers appear in differential topology, coding theory, theoretical physics, and linear algebra. For example, \(\rho (m)\) counts the maximum size of a linear subspace of the real \(m\times m\) matrices for which each nonzero matrix is a product of an orthogonal matrix and a scalar matrix. In the paper the author proves the following result: Let \(n\in \mathbb N\). The maximum dimension of a space \(\mathcal{V}\) of \((2n+1)\times (2n+1)\) real skew-symmetric matrices where every \(A\in \mathcal{V}\) is orthogonally similar to a matrix of the form \[ \biggl[\begin{matrix} 0 & \lambda _{A} \\ -\lambda _{A} & 0 \end{matrix}\biggr] \oplus \dots\oplus \biggl[\begin{matrix} 0 & \lambda _{A} \\ -\lambda _{A} & 0\end{matrix}\biggr] \oplus 0_{1},\tag{1} \] with nonzero \(\lambda _{A}\in \mathbb R\), equals \(\rho (2n)-1\) if \(n\) is even, and \(\rho (2n+2)-1\) if \(n\) is odd. The paper has two chapters. The result is stated in the first chapter. The proof of the result is presented in the second chapter. It is based on several lemmas. A matrix \(A\) is orthogonally similar to a matrix of the form ({1}) if and only if \(A\) is a real skew-symmetric matrix of corank \(1\) satisfying \(A^{3}=\lambda A\) for some real \(\lambda <0\). Note to readers: It is mistakenly written in the abstract of the paper that this scalar \(\lambda \) is positive however this mistake is only of a technical nature, namely it does not occur later in the paper.
reviewed by: Janko Marovt
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 6125871
zbMATH Keywords: skew-symmetric matrix
zbMATH Keywords: maximal linear subspace
zbMATH Keywords: Radon-Hurwitz number
zbMATH Keywords: matrix space
zbMATH Keywords: orthogonal similarity
MaRDI profile type: Publication
full work available at URL: j.laa.2012.10.007
OpenAlex ID: W165697298
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DOI: 10.1016/J.LAA.2012.10.007
description: scientific article; zbMATH DE number 6125871