= Reciprocal spectrum of a symplectic matrix
{title2=$\chi_M(x)=x^{2m}\chi_M(x^{-1})$}
For a <symplectic matrix> $M$ of size $2m$ with determinant one, the defining identity $M^TJM=J$ gives $M^{-1}=J^{-1}M^TJ$. Since similarity and transposition preserve the <characteristic polynomial>, $\chi_{M^{-1}}=\chi_M$. Factoring $xI-M=M(xM^{-1}-I)$ then proves the displayed reciprocal identity for $x\ne0$. If $\chi_M(z)=(z-\lambda)^dR(z)$ with $R(\lambda)\ne0$, this identity gives $\chi_M(x)=x^{2m-d}(1-\lambda x)^dR(1/x)$ near $x=\lambda^{-1}$. Thus reciprocal <eigenvalues> have the same <algebraic multiplicity>. The fixed points $\pm1$ require the stronger parity argument in <even multiplicities at fixed points of reciprocal root pairing>.
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