A Hessenberg–Jacobi isospectral flow
Tóm tắt
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, 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.
Tài liệu tham khảo
Lax P.: Integrals of nonlinear equations of evolution and solitary waves Comm. Pure Appl. Math. 21, 467–490 (1968)
Flaschka H.: The Toda lattice, I. Phys. Rev. B 9, 1924–1925 (1974)
Moser, J.: Finitely many mass points on the line under the influence of an exponential potential an integrable system. In: Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 467–497. Lecture Notes in Physics, vol. 38. Springer, Berlin (1975)
Deift P., Li L.-C., Tomei C.: Matrix factorizations and integrable systems. Commun. Pure Appl. Math. 42(4), 443–521 (1989)
Symes W.: Hamiltonian group actions and integrable systems. Phys. D 1, 339–374 (1980)
Perelomov, A.M.: Integrable Systems of Classical Mechanics and Lie Algebras, vol. 1. Birkhäuser, Basel (1990)
Brockett R.: Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems. Linear Algebra Appl. 146, 79–91 (1991)
Chu M.: Linear algebra algorithms as dynamical systems. Acta Numer. 17, 1–86 (2008)
Helmke, U., Moore, J.: Optimization and Dynamical Systems. Springer, Berlin (1994)
Tomei C.: The topology of manifolds of isospectral tridiagonal matrices. Duke Math. J. 51, 981–996 (1984)
Leite R., Saldanha N., Tomei C.: An atlas for tridiagonal isospectral manifolds. Linear Algebra Appl. 429(1), 387–402 (2008)
Ebenbauer, C., Arsie, A.: On an eigenflow equation and its structure preserving properties. Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. In: Proceedings of the 48th IEEE conference on publication, pp. 7491–7496 (2009)
Lax, P.: Linear Algebra and Its Applications, Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts (2007)
Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Ryashko L.B., Shnol E.E.: On exponentially attracting invariant manifolds of ODEs. Nonlinearity 16, 147–160 (2003)
Vinter, R.: Optimal Control. Modern Birkhäuser Classics (2010)
Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)