A normalizing isospectral flow on complex Hessenberg matrices

We study an isospectral flow (Lax flow) that provides an explicit deformation from upper Hessenberg complex matrices to normal matrices, extending to the complex case and to the case of normal matrices the results of [2]. The Lax flow is given bydAdt=[[A†,A]du,A], where brackets indicate the usual matrix commutator, [A,B]:=AB−BA[A,B]:=AB−BA, A†A† is the conjugate transpose of A   and the matrix [A†,A]du[A†,A]du is the matrix equal to [A†,A][A†,A] along diagonal and upper triangular entries and zero below diagonal. We prove that if the initial condition A0A0 is an upper Hessenberg matrix with simple spectrum, then limt→+∞⁡A(t)limt→+∞⁡A(t) exists and it is a normal upper Hessenberg matrix isospectral to A0A0 and if the spectrum of A0A0 is contained in a line in the complex plane, then the ω-limit set is actually a tridiagonal normal matrix. Furthermore, we show that this flow is also the solution of an infinite time horizon optimal control problem and we prove that it can be used to construct even dimensional real skew-symmetric tridiagonal matrices with given simple spectrum, and with given signs pattern for the codiagonal elements.