  Associated with every $2n\times 2n$ real positive definite matrix $A,$ there
exist $n$ positive numbers called the symplectic eigenvalues of $A,$ and a
basis of $\mathbb{R}^{2n}$ called the symplectic eigenbasis of $A$
corresponding to these numbers. In this paper, we discuss the differentiability
(analyticity) of the symplectic eigenvalues and corresponding symplectic
eigenbasis for differentiable (analytic) map $t\mapsto A(t),$ and compute their
derivatives. We then derive an analogue of Lidskii's theorem for symplectic
eigenvalues as an application.
