Linear Hamiltonian systems with periodic coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. I will show that the solution map of a linear Hamiltonian system with periodic coefficients can be parameterized by an envelope matrix, which has a clear physical meaning and satisfies a nonlinear envelope matrix equation. The Hamiltonian system is stable if and only if the envelope matrix equation admits a periodic (matched) solution. The mathematical devices utilized in this theoretical development with significant physical implications are time-dependent canonical transformations, normal forms and Iwasawa decomposition of symplectic matrices.