The following result is attributed to J. Williamson: Every real, symmetric, and positive definite matrix A of even order n = 2m can be brought to diagonal form by a congruence transformation with symplectic matrix. The diagonal entries of this form are invariants of congruence transformations performed with A, and they are called the symplectic eigenvalues of this matrix. This short paper proves an analogous fact concerning (complex) skew-symmetric matrices and transformations belonging to a different group, namely, the group of pseudo-orthogonal matrices.
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J. Williamson, “On the algebraic problem concerning the normal form of linear dynamical systems,” Amer. J. Math., 58, 141–163 (1936).
Kh. D Ikramov, “On the symplectic eigenvalues of positive definite matrices,” Vestn. Mosk. Univ., Ser. 15, Vychisl. Mat. Kibern., No. 1, 3–6 (2018).
A. I. Mal’tsev, Fundamentals of Linear Algebra [in Russian], Nauka, Moscow (1970).
Kh. D. Ikramov, “On the singular values and polar decomposition of an operator in a space with bilinear metric,” Zh. Vychisl. Mat. Mat. Fiz., 28, 127–129 (1988).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 472, 2018, pp. 92–97.
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Ikramov, K.D. Pseudo-Orthogonal Eigenvalues of Skew-Symmetric Matrices. J Math Sci 240, 765–768 (2019). https://doi.org/10.1007/s10958-019-04393-9
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DOI: https://doi.org/10.1007/s10958-019-04393-9