I want to prove that any symmetric positive definite symplectic matrix, $A$, and any real number $\alpha >0$, also $A^{\alpha} \in \operatorname{Sp}(2n)$.
I was given a hint to decompose $\mathbb{R}^{2n}$ into direct sum of $V_{\lambda}$ the eigenspaces, where $\lambda \in \sigma(A)$, and then use the characterization of the eigenvalues of $A$.
I must say, that I feel totally clueless here. :-(