Spaces of skew-symmetric matrices satisfying \(A^3=\lambda A\)
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.
scientific article; zbMATH DE number 6125871