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PERTURBATIONOFPURELYIMAGINARYEIGENVALUESOFHAMILTONIANMATRICESUNDERSTRUCTUREDPERTURBATIONS ELA

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ELA

PERTURBATION OF PURELY IMAGINARY EIGENVALUES OF HAMILTONIAN MATRICES UNDER STRUCTURED

PERTURBATIONS

∗

VOLKER MEHRMANN† AND HONGGUO XU‡

Abstract. The perturbation theory for purely imaginary eigenvaluesof Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations is discussed. It is shown that there is a substantial difference in the behavior under these perturbations. The perturbation of real eigenvalues of real skew-Hamiltonian matrices under structured perturbations is discussed as well and these results are used to analyze the properties of the URV method for computing the eigenvalues of Hamiltonian matrices.

Key words. Hamiltonian matrix, Skew-Hamiltonian matrix, Symplectic matrix, Structured perturbation, Invariant subspace, Purely imaginary eigenvalues, Passive system, Robust control, Gyroscopic system.

AMS subject classifications.15A18, 15A57, 65F15, 65F35.

∗Received by the editorsSeptember 30, 2007. Accepted for publication April 17, 2008. Handling Editor: Harm Bart.

†Institut f¨ur Mathematik, TU Berlin, Str. des17. Juni 136, D-10623 Berlin, FRG ([email protected]). Partially supported byDeutsche Forschungsgemeinschaft, through the DFG Research CenterMatheonMathematics for Key Technologiesin Berlin.

‡Department of Mathematics, University of Kansas, Lawrence, KS 44045, USA. Partially sup- ported bythe University of Kansas General Research Fund allocation # 2301717and byDeutsche Forschungsgemeinschaftthrough the DFG Research CenterMatheonMathematics for key technolo- giesin Berlin. Part of the work was done while this author was visiting TU Berlin whose hospitality isgratefully acknowledged.

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 17, pp. 234-257, April 2008

http://math.technion.ac.il/iic/ela

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