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A Hessenberg–Jacobi isospectral flow

  • Alessandro Arsie
  • Christian Ebenbauer
Article

Abstract

In this paper we introduce an isospectral flow (Lax flow) that deforms real Hessenberg matrices to Jacobi matrices isospectrally. The Lax flow is given by
$$\frac{dA}{dt} = [[A^T, A]_{du}, A],$$
where brackets indicate the usual matrix commutator, [A, B] : = ABBA, A T is the transpose of A and the matrix [A T , A] du is the matrix equal to [A T , A] along diagonal and upper triangular entries and zero below diagonal. We prove that if the initial condition A 0 is upper Hessenberg with simple spectrum and subdiagonal elements different from zero, then \({\lim_{t\rightarrow +\infty}A(t)}\) exists, it is a tridiagonal symmetric matrix isospectral to A 0 and it has the same sign pattern in the codiagonal elements as the initial condition A 0. Moreover we prove that the rate of convergence is exponential and that this system is the solution of an infinite horizon optimal control problem. Some simulations are provided to highlight some aspects of this nonlinear system and to provide possible extensions to its applicability.

Mathematics Subject Classification (2010)

34C40 34D05 34C45 15B35 15A29 

Keywords

Isospectral flow Jacobi matrices Omega-limit set Differential equations on manifolds 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of ToledoToledoUSA
  2. 2.Institute for Systems Theory and Automatic ControlUniversity of StuttgartStuttgartGermany

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