A normalizing isospectral flow on complex Hessenberg matrices

作者：Arsie Alessandro^*; Pokharel Krishna

来源：Journal of Mathematical Analysis and Applications, 2015, 432(2): 787-805.

DOI：10.1016/j.jmaa.2015.07.020

• Isospectral flow
• Normal matrices
• Omega-limit set
• Differential equations on manifolds

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 by
dA/dt = [[A(dagger), A](du), A],
where brackets indicate the usual matrix commutator, [A, B] := AB - BA, A(dagger) is the conjugate transpose of A and the matrix [A(dagger), A](du) is the matrix equal to [A(dagger), A] along diagonal and upper triangular entries and zero below diagonal. We prove that if the initial condition A(0) is an upper Hessenberg matrix with simple spectrum, then lim(t ->+infinity) A(t) exists and it is a normal upper Hessenberg matrix isospectral to Ao and if the spectrum of A(0) is contained in a line in the complex plane, then the omega-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.

• 出版日期2015-12-15