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Isospectral Flows on a Class of Finite-Dimensional Jacobi Matrices

Author(s):

T. Sutter
Conference/Journal:

Semester Thesis, FS 11 (10083)
Abstract:

In this report a new matrix-valued isospectral o.d.e. is presented, which asymptotically block-diagonalizes a finite-dimensional zero-diagonal Jacobi matrix employed as its initial condition. This differential equation is closely related to the one introduced by Kac and van Moerbeke, although, our approach to prove the remarkable properties of this o.d.e. differs from the techniques due to Kac and van Moerbeke. We show that our o.d.e. can be represented as a double bracket differential equation similar to the one studied by Brockett in his 1991 seminal work. Simulation results are provided, which suggest that on the set of zero-diagonal Jacobi matrices the solutions to our o.d.e. have considerably faster convergence compared to that of Brockett. Furthermore, the o.d.e. presented can be expanded to the set of real symmetric matrices while many properties are preserved. Our work is motivated by results (e.g., as in Brockett’s 1991 work) suggesting that tasks such as diagonalizing matrices, sorting a list, and solving linear programming optimization problems, traditionally associated with computer science algorithms, can be answered by means of smooth dynamical systems.

Year:

2011
Type of Publication:

(13)Semester/Bachelor Thesis
Supervisor:

D. Chatterjee

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% Autogenerated BibTeX entry
@PhdThesis { Xxx:2011:IFA_4111
}
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