Let Ip,q = Ip ⊕−Iq. Pseudounitary eigenvalues of a positive definite matrix A are the moduli of conventional eigenvalues of the matrix Ip,qA. They are invariants of pseudounitary *-congruences performed with A. For a fixed n = p+q, the sum of the squares σp,q of these numbers is a function of the parameter p. In general, its values for different p can differ very significantly. However, if A is the tridiagonal Toeplitz matrix with an entry a ≥ 2 on the principal diagonal and the entry −1 on the two neighboring diagonals, then σp,q has the same value for all p. This nontrivial fact is explained in the paper.
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Kh. D. Ikramov and A. M. Nazari, “From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues,” Comput. Math. Computer Model. Appl., 1, No. 1, 17–20 (2022).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 514, 2022, pp. 55–60.
Translated by Kh. D. Ikramov.
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Ikramov, K.D. On a Nontrivial Situation Concerning the Pseudounitary Eigenvalues of a Positive Definite Matrix. J Math Sci 272, 519–522 (2023). https://doi.org/10.1007/s10958-023-06445-7
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DOI: https://doi.org/10.1007/s10958-023-06445-7