無中文摘要

In applications a symplectic matrix is often required to be partitioned with a nonsingular block. By applying the complementary bases theorem of Dopico and Johnson in [3], we can rearrange a symplectic matrix with a swap matrix to obtain a nonsingular block. We classify symplectic matrices with corresponding swap matrices. Moreover, a rearrangement of symplectic pair by Mehrmann and Poloni in [8] merges a regular symplectic pair into a symplectic matrix. Therefore
we can classify regular symplectic pairs with similar approach.

References
[1] R. Abraham and J. Marsden. Foundations of Mechanics, second ed. Addison-Wesley, Reading, 1978.
[2] V.I. Arnold. Mathematical Methods in Classical Mechanics. Springer-Verlag, Berlin, 1978.
[3] Froiln M. Dopico and Charles R. Johnson. Complementary bases in symplectic matrices and a proof that their determinant is one. Linear Algebra and its Applications,
419(2):772-778, 2006.
[4] H. Fassbender. Symplectic Methods for the Symplectic Eigenproblem. Springer-Verlag, Berlin, 2000.
[5] D.S. Mackey and N. Mackey. On the Determinant of Symplectic Matrices, Numerical Analysis Report No. 422. Manchester Centre for Computational Mathematics, Manchester, England, 2003.
[6] D. McDuff and D. Salamon. Introduction to Symplectic Topology. Clarendon Press, Oxford, 1995.
[7] V. Mehrmann. The autonomous linear quadratic control problem - theory and numerical solutions, lecture notes in control and information sciences, vol. 163, 1991.
[8] Volker Mehrmann and Federico Poloni. Doubling algorithms with permuted lagrangian graph bases. SIAM Journal on Matrix Analysis and Applications, 33(3):780-805, 2012.