Theorem 1   The characteristic polynomial of a symplectic matrix is a reciprocal polynomial.

Proof. Let $A$ be the symplectic matrix, and let $p(\lambda) = \det(A-\lambda I)$ be its characteristic polynomial. We wish to prove that $$ p(\lambda) = \pm \lambda^n p(1/\lambda). $$ By definition, $AJA^T=J$ where $J$ is the matrix

Since $A$ and $J$ are symplectic matrices, their determinants are $1$ , and \begin{eqnarray*} p(\lambda) &=& \det (AJ - \lambda J) \\ &=&\det (AJ - \lambda AJA^T) \\ &=&\det (-\lambda A) \det (J) \det (-\frac{1}{\lambda} J + JA^T) \\ &=&\pm \lambda^n \det (A-\frac{1}{\lambda} I ). \end{eqnarray*}as claimed.