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 Zh. Vychisl. Mat. Mat. Fiz., 2012, Volume 52, Number 1, Pages 4–7 (Mi zvmmf9632)

Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm

Kh. D. Ikramov

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992 Russia

Abstract: Takagi’s decomposition is an analog (for complex symmetric matrices and for unitary similarities replaced by unitary congruences) of the eigenvalue decomposition of Hermitian matrices. It is shown that, if a complex matrix is not only symmetric but is also unitary, then its Takagi decomposition can be found by quadratic radicals, that is, by means of a finite algorithm that involves arithmetic operations and quadratic radicals. A similar fact is valid for the eigenvalue decomposition of reflections, which are Hermitian unitary matrices.

Key words: unitary matrices, symmetric matrices, Takagi’s decomposition, solvability by quadratic radicals

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English version:
Computational Mathematics and Mathematical Physics, 2012, 52:1, 1–3

Bibliographic databases:

UDC: 519.61

Citation: Kh. D. Ikramov, “Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm”, Zh. Vychisl. Mat. Mat. Fiz., 52:1 (2012), 4–7; Comput. Math. Math. Phys., 52:1 (2012), 1–3

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Yu. O. Vorontsov, Kh. D. Ikramov, “Numerical algorithm for solving quadratic matrix equations of a certain class”, Comput. Math. Math. Phys., 54:11 (2014), 1643–1646
2. Marcellan F., Shayanfar N., “OPUC, CMV Matrices and Perturbations of Measures Supported on the Unit Circle”, Linear Alg. Appl., 485 (2015), 305–344
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