Geodesic
Effective Algorithms for Decomplexifying a Matrix by Unitary Similarities or Congruences
Matematičeskie zametki, Tome 92 (2012) no. 6, pp. 856-863.

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It is required to verify whether a given complex $n\times n$ matrix $A$ can be made real by a similarity or a congruence transformation. Algorithms for solving these two problems are proposed and justified under the additional assumption that $A$ is irreducible in the former case and $A_L=\overline AA$ is irreducible in the latter case. The irreducibility of a square complex matrix means that no unitary similarity transformation converts this matrix into a direct sum of smaller matrices. The proposed algorithms are effective in the sense that their implementation requires a finite number of arithmetic operations.
Keywords: unitary similarity, unitary congruence, irreducible matrix, consimilarity.
Mots-clés : polar decomposition 
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Kh. D. Ikramov. Effective Algorithms for Decomplexifying a Matrix by Unitary Similarities or Congruences. Matematičeskie zametki, Tome 92 (2012) no. 6, pp. 856-863. http://geodesic.mathdoc.fr/item/MZM_2012_92_6_a5/

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