The continuous-time Rayleigh quotient flow on the sphere
type: Item
instance of: scholarly article
title: The continuous-time Rayleigh quotient flow on the sphere
zbMATH Open document ID: 1030.65025
DOI: 10.1016/S0024-3795(02)00730-9
published in: Linear Algebra and its Applications
publication date: 2003-07-30T00:00:00Z
review text: Consider the eigenvalue problem for a symmetric matrix. The Rayleigh quotient can be defined by \(r_{A}(x)=x^{T}Ax\) where \(x\) is on the sphere \( x^{T}x=1,\) \ \(x\in \mathbb{R}^{n}.\) Using the projection onto the tangent space, the continuous time Rayleigh quotient flow is generated by \(\dot{x}=\pi _{x}\left( A-r_{A}(x)I_{n}\right) ^{-1}x\) where \(x(0)\in S^{n-1}.\) Thus the vector field is defined except when \(r_{A}(x)\) is an eigenvalue. The phase portrait is analyzed and it is shown that the flow of a nondegenerate solution visits each of the eigenvectors in an ascending order.
reviewed by: Amin Boumenir
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 1956810
zbMATH Keywords: Rayleigh quotient flow
zbMATH Keywords: inverse iteration
zbMATH Keywords: eigenvalue
zbMATH Keywords: symmetric matrix
zbMATH Keywords: phase portrait
zbMATH Keywords: eigenvectors
author: Robert Mahony
author: P.-A. Absil
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1956810