Diagonals of rotation matrices
An orthogonal matrix is a real \(n\times n\) matrix \(A\) with \(A^ T=A^{- 1}\). An \(n\times n\) doubly stochastic matrix \(Q=(q_{ij})\) is orthostochastic if there exists an orthogonal matrix \(A=(a_{ij})\) such that \(q_{ij}=a^ 2_{ij}\) for all \(i,j\). \textit{A. Horn} [Amer. J. Math. 76, 620-630 (1954; Zbl 0055.246)] has shown that if \(x,y\in\mathbb{R}^ n\) and \(y\) belongs to the convex hull of all permutations \((x_{\sigma(1)},\ldots,x_{\sigma(n)})^ T\) of \(x=(x_ 1,\ldots,x_ n)^ T\), then for some orthostochastic matrix \(Q\), \(y=Qx\). Without using the Horn result the author shows that the set \(D\) of all diagonals of rotations of order \(n\) is the convex hull of those points \((\pm 1,\ldots,\pm 1)\) of which an even number (possibly zero) of coordinates are \(-1\).
scientific article; zbMATH DE number 205212