| This thesis provides a quaternionic representation of real symplectic matrices in four dimensions analogous to that for the orthogonal group, and explicit techniques to compute the minimal polynomials of a wide variety of structured matrices that include Hamiltonian and orthogonal matrices. It also provides a technique to compute this quaternionic representation for symplectic matrices from the entries of the matrix being represented. In the process, it also shows how to compute the polar decomposition of a 4 x 4 symplectic matrix without any recourse to spectral calculations. Applications of this work to quantum optics and dynamic systems in different aspects have been discussed here. Extensions to higher dimensions for minimal polynomials in Clifford algebras are also discussed. |
