Canonical forms for Hamiltonian and symplectic matrices and pencils

Wen-Wei Lin; Volker Mehrmann; Hongguo Xu

doi:10.1016/S0024-3795(99)00191-3

Canonical forms for Hamiltonian and symplectic matrices and pencils

Wen-Wei Lin, Volker Mehrmann^*, Hongguo Xu

^*Corresponding author for this work

Department of Applied Mathematics

Research output: Contribution to journal › Article › peer-review

86 Scopus citations

Abstract

We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho (On Schur type decompositions for Hamiltonian and symplectic pencils, Technical Report, Institute of Applied Mathematics, National Tsing Hua University, Taiwan, 1990) and simplify the proofs presented there.

Original language	English
Pages (from-to)	469-533
Number of pages	65
Journal	Linear Algebra and Its Applications
Volume	302-303
DOIs	https://doi.org/10.1016/S0024-3795(99)00191-3
State	Published - 1 Dec 1999

Keywords

Algebraic Riccati equation
Eigenvalue problem
Hamiltonian pencil (matrix)
Jordan canonical form
Kronecker canonical form
Linear quadratic control
Symplectic pencil (matrix)

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abstract = "We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho (On Schur type decompositions for Hamiltonian and symplectic pencils, Technical Report, Institute of Applied Mathematics, National Tsing Hua University, Taiwan, 1990) and simplify the proofs presented there.",

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AU - Lin, Wen-Wei

AU - Mehrmann, Volker

AU - Xu, Hongguo

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N2 - We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho (On Schur type decompositions for Hamiltonian and symplectic pencils, Technical Report, Institute of Applied Mathematics, National Tsing Hua University, Taiwan, 1990) and simplify the proofs presented there.

AB - We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho (On Schur type decompositions for Hamiltonian and symplectic pencils, Technical Report, Institute of Applied Mathematics, National Tsing Hua University, Taiwan, 1990) and simplify the proofs presented there.

KW - Algebraic Riccati equation

KW - Eigenvalue problem

KW - Hamiltonian pencil (matrix)

KW - Jordan canonical form

KW - Kronecker canonical form

KW - Linear quadratic control

KW - Symplectic pencil (matrix)

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JO - Linear Algebra and Its Applications

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