  For every $2n \times 2n$ positive definite matrix $A$ there are $n$ positive
numbers $d_1(A) \leq \ldots \leq d_n(A)$ associated with $A$ called the
symplectic eigenvalues of $A.$ It is known that $d_m$ are continuous functions
of $A$ but are not differentiable in general. In this paper, we show that the
directional derivative of $d_m$ exists and derive its expression. We also
discuss various subdifferential properties of $d_m$ such as Clarke and
Michel-Penot subdifferentials.
