Geodesic
Symplectic eigenvalues and singular values of symmetric matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIV, Tome 504 (2021), pp. 61-69
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Williamson's theorem on the symplectic eigenvalues of symmetric positive definite matrices is interpreted in terms of special operators of the real symplectic space and their spectra. A relation connecting the conventional and symplectic eigenvalues of a given matrix is derived. 
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Kh. D. Ikramov; A. M. Nazari. Symplectic eigenvalues and singular values of symmetric matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIV, Tome 504 (2021), pp. 61-69. http://geodesic.mathdoc.fr/item/ZNSL_2021_504_a4/

[1] J. Williamson, “On the algebraic problem concerning the normal form of linear dynamical systems”, Amer. J. Math., 58:1 (1936), 141–163 | DOI | MR

[2] Kh. D. Ikramov, Kh. Fassbender, “O proizvedenii dvukh kosogamiltonovykh ili dvukh kososimmetrichnykh matrits”, Zap. nauchn. semin. POMI, 359, 2008, 45–51

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