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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
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/
@article{MZM_2010_87_6_a4,
     author = {Kh. D. Ikramov},
     title = {On {Complex} {Matrices} that {Are} {Unitarily} {Similar} to {Real} {Matrices}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {840--847},
     year = {2010},
     volume = {87},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_6_a4/}
}
TY  - JOUR
AU  - Kh. D. Ikramov
TI  - On Complex Matrices that Are Unitarily Similar to Real Matrices
JO  - Matematičeskie zametki
PY  - 2010
SP  - 840
EP  - 847
VL  - 87
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MZM_2010_87_6_a4/
LA  - ru
ID  - MZM_2010_87_6_a4
ER  - 
%0 Journal Article
%A Kh. D. Ikramov
%T On Complex Matrices that Are Unitarily Similar to Real Matrices
%J Matematičeskie zametki
%D 2010
%P 840-847
%V 87
%N 6
%U http://geodesic.mathdoc.fr/item/MZM_2010_87_6_a4/
%G ru
%F MZM_2010_87_6_a4

[1] G. E. Shilov, Matematicheskii analiz. Konechnomernye lineinye prostranstva, Nauka, M., 1969 | MR | Zbl

[2] R. Khorn, Ch. Dzhonson, Matrichnyi analiz, Mir, M., 1989 | MR | Zbl

[3] C. Pearcy, “A complete set of unitary invariants for operators generating finite $W^*$-algebras of type I”, Pacific J. Math., 12 (1962), 1405–1416 | MR | Zbl

[4] Kh. D. Ikramov, “O konechnom ratsionalnom kriterii neprivodimosti matrits”, Vestn. Mosk. un-ta. Ser. 15. Vychislit. matem., kibernet., 2007, no. 3, 16–18 | MR | Zbl

[5] Kh. D. Ikramov, “O kompleksnykh matritsakh, oveschestvlyaemykh posredstvom unitarnogo podobiya”, Dokl. RAN, 430:1 (2010), 15–17