THE EIGENVECTORS OF A REAL SYMMETRIC MATRIX ARE ASYMPTOTICALLY STABLE FOR SOME DIFFERENTIAL EQUATION,

Abstract

Let A be a real symmetric n x n matrix. For each real unit vector x one computes numbers mu = mu(x) and sigma = sigma(x), which have the property that (mu + sigma, mu - sigma) contains an eigenvalue of A. An autonomous differential equation is established, dependent on A, which admits asymptotically stable solutions of the form, x = eigenvector of A. This is achieved by noting sigma squared (x) is a Liapunov function, and tends monotonically to zero along solutions of the differential equation. The set of critical points of sigma squared (x) are shown to comprise a finite union of products of spheres. (Author)

Document Details

Document Type
Technical Report
Publication Date
Jul 01, 1970
Accession Number
AD0708502

Entities

People

  • Stephen H. Saperstone

Organizations

  • Center for Naval Analyses

Tags

DTIC Thesaurus Topics

  • Differential Equations
  • Eigenvalues
  • Eigenvectors
  • Equations
  • Mathematical Analysis
  • Mathematics
  • Real Variables

Fields of Study

  • Mathematics

Readers

  • Analytical Mechanics
  • Calculus or Mathematical Analysis
  • Linear Algebra