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Symplectic Matrix, Eigenvalues and Multiplicity

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A 2n x 2n M is symplectic if where J is the (also 2n x 2n) matrix .
Prove that if is an eigenvalue of M , then so is , and that these have the same multiplicity.
Show furthermore that, if are eigenvalues of M, and , then the corresponding eigenvectors have the property that

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Symplectic Matrix, Eigenvalues and Multiplicity are investigated. The solution is detailed and well presented.

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