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Let's consider two symmetric real matrices $A$ and $B$ of dimension $2N$ and with the same (algebraic) eigenvalues, possibly degenerate.

Is there a simple criterion to tell whether there exists or not a symplectic orthogonal matrix $S$ that transforms $A$ into $B$ ? $$B=S^\top A S $$

Can I build explicitly such a matrix ?

Thanks, Olivier

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