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symplectic matrix

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Definition
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greetings.
I learn from various sources that the eigenvalues of a symplectic matrix occur in reciprocal pairs. However, i also find that there are some matrices that, while their eigenvalues also occur in reciprocal pairs, they are not symplectic. Therefore,I want to know what are the properties of a matrix that has paired reciprocal eigenvalues.

thanks to everyone who answered my question

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