Geodesic
Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 1, pp. 4-7

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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. 
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Kh. D. Ikramov. Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 1, pp. 4-7. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_1_a1/