Geodesic
On Complex Matrices that Are Unitarily Similar to Real Matrices
Matematičeskie zametki, Tome 87 (2010) no. 6, pp. 840-847.

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There are well-known conditions ensuring that a complex $n\times n$ matrix $A$ can be converted by a similarity transformation into a real matrix. Is it possible to realize this conversion via unitary similarity rather than a general one? The following answer to this question is given in this paper: A matrix $A\in M_n(\mathbb C)$ can be made real by a unitary similarity transformation if and only if $A$ and $\overline A$ are unitarily similar and the matrix $P$ transforming $A$ into $\overline A$ can be chosen unitary and symmetric at the same time. Effective ways for verifying this criterion are discussed.
Mots-clés : complex matrix
Keywords: unitary similarity transformation, irreducible matrix, block quaternion, Jordan block, Specht's criterion. 
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Kh. D. Ikramov. On Complex Matrices that Are Unitarily Similar to Real Matrices. Matematičeskie zametki, Tome 87 (2010) no. 6, pp. 840-847. http://geodesic.mathdoc.fr/item/MZM_2010_87_6_a4/

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