Diagonals of rotation matrices
type: Item
instance of: scholarly article
title: Diagonals of rotation matrices
zbMATH Open document ID: 0778.15010
DOI: 10.1016/0024-3795(93)90293-W
author: Alberto Borobia
published in: Linear Algebra and its Applications
publication date: 1994-01-03T00:00:00Z
review text: 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 \(-1\).
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 205212
zbMATH Keywords: orthogonal matrix
zbMATH Keywords: doubly stochastic matrix
zbMATH Keywords: orthostochastic matrix
zbMATH Keywords: convex hull
reviewed by: Richard Sinkhorn
MaRDI profile type: Publication
cites work: Q5812325
cites work: Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix
cites work: Results and problems in the theory of doubly-stochastic matrices
cites work: Inequalities and existence theorems in the theory of matrices
full work available at URL: 0024-3795(93)90293-w
OpenAlex ID: W2006562178
description: scientific article; zbMATH DE number 205212