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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     author = {Kh. D. Ikramov},
     title = {Effective {Algorithms} for {Decomplexifying} a {Matrix} by {Unitary} {Similarities} or {Congruences}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {856--863},
     publisher = {mathdoc},
     volume = {92},
     number = {6},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_92_6_a5/}
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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/