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I have 2 symplectic matrices $X_{1},X_{2} \in \mathbb{R}^{2n\times2n}$. The matrix $X=X_{1} \cdot X_{2}$ is also symplectic.

Question: Are there any theorems which allow me to express eigenvalues of $X$ if I know eigenvalues of each $X_i$?

P.S. Actually I am interested in some numerical algorithms. I cannot mupliply these matrices direcly because time-to-time this leads to overflow (or accuracy loss) problems. Maybe there are some iterative procedures? Some helpful decomposition of matrix?

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  • $\begingroup$ It would surprise me if the answer is yes. There are basically no results on expressing eigenvalues of products in terms of the eigenvalues of the multiplicand matrices (apart from zero eigenvalues, which your matrices do not have since symplectic implies nonsingular). Multiplying matrices with rational eigenvalues can give matrices with irrational (even non-algebraic) eigenvalues, so finding an exact result seems implausible. $\endgroup$ – Federico Poloni Dec 18 '15 at 13:08
  • $\begingroup$ Thank you. I know that there are no general theorems for a product of matrices, but I thought symplecticity property could be helpful here. Anyway my problem is not so global, so I clarified my question a bit. $\endgroup$ – Maksim Surov Dec 18 '15 at 13:23
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Answering to your edit: there are numerical methods to compute the eigenvalues of a matrix product without forming it first. This has stability advantages with respect to forming the product explicitly. There is an excellent review by D.S. Watkins, Product eigenvalue problems, 2005, Siam review. As far as I know there is nothing relative to symplectic matrices specifically.

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  • $\begingroup$ Good paper, I will take a look. $\endgroup$ – Maksim Surov Dec 18 '15 at 14:23

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