A Hessenberg–Jacobi isospectral flow

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] : = AB−BA, AT is the transpose of A and the matrix [AT, A]du is the matrix equal to [AT, A] along diagonal and upper triangular entries and zero below diagonal. We prove that if the initial condition A0 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 A0 and it has the same sign pattern in the codiagonal elements as the initial condition A0. 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.