Problem-symplectic matrix and eigenvalues and multiplicity

Question

Question:

Symplectic Matrix, Eigenvalues and Multiplicity

A 2n x 2n M is symplectic if M^TJM = J where J is the (also 2n x 2n) matrix ( 0   -I ).
                                                                                                                  ( I    0 )

Prove that if λ is an eigenvalue of M , then so is λ^-1 , and that these have the same multiplicity.

Show furthermore that if λ and µ are eigenvalues of M, and λ ≠ µ ,then the corresponding eigenvectors gλ, gµ have the property that

g^T µ^J g^λ = 0