Normal forms of symplectic matrices (Q1578863)

A basic way to study the problems related to a class of matrices is to reduce these matrices via suitable coordinate changes which preserve the basic property of the problem in consideration to a smaller family which is called the normal forms. The authors define the complete set of normal forms for real symplectic matrices possessing eigenvalues on the unit circle of the complex plane. The normal forms of symplectic matrices possessing eigenvalues away from the unit circle were considered by \textit{J. Han} and \textit{Y. Long} [Normal forms of symplectic matrices II. Research Report, Nankai Inst. of Math. Nankai Univ. (1996)]. Together with this paper the results of the present paper contain all cases for real symplectic matrices. The new interest on such normal forms comes from the study of the topological structures of the symplectic groups and their subsets, which forms the basis for defining and studying the Maslov-type index theory and its applications to linear Hamiltonian systems with periodic coefficients, and the periodic solutions of nonlinear Hamiltonian systems with Hamiltonians periodic in time [see for example \textit{Y. Long}, Pac. J. Math. 187, No. 1, 113-149 (1999; Zbl 0924.58024)].