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Matematicheskie Zametki, 2010, Volume 87, Issue 6, Pages 840–847
DOI: https://doi.org/10.4213/mzm8532 (Mi mzm8532) 		 
This article is cited in 9 scientific papers (total in 9 papers)

On Complex Matrices that Are Unitarily Similar to Real Matrices

Kh. D. Ikramov

M. V. Lomonosov Moscow State University
Full-text PDF (437 kB) Citations (9)
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DOI: https://doi.org/10.4213/mzm8532
Abstract: There are well-known conditions ensuring that a complex n×nn×n matrix AA 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∈Mn(C) can be made real by a unitary similarity transformation if and only if A and ¯A are unitarily similar and the matrix P transforming A into ¯A can be chosen unitary and symmetric at the same time. Effective ways for verifying this criterion are discussed.
Keywords: complex matrix, unitary similarity transformation, irreducible matrix, block quaternion, Jordan block, Specht's criterion.
Received: 03.08.2009
Revised: 01.12.2009
English version:
Mathematical Notes, 2010, Volume 87, Issue 6, Pages 821–827
DOI: https://doi.org/10.1134/S0001434610050214
Bibliographic databases:
Document Type: Article
UDC: 512.643
Language: Russian
Citation: Kh. D. Ikramov, “On Complex Matrices that Are Unitarily Similar to Real Matrices”, Mat. Zametki, 87:6 (2010), 840–847; Math. Notes, 87:6 (2010), 821–827
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/mzm8532
  • https://doi.org/10.4213/mzm8532
  • https://www.mathnet.ru/eng/mzm/v87/i6/p840
    • This publication is cited in the following 9 articles:
    1. Kh. D. Ikramov, “Klassy psevdoperestanovochnosti kompleksnykh matrits i ikh oveschestvlenie”, Sib. zhurn. vychisl. matem., 26:2 (2023), 199–203  mathnet  crossref
    2. Kh. D. Ikramov, “Pseudo-Commutation Classes of Complex Matrices and Their Decomplexification”, Numer. Analys. Appl., 16:2 (2023), 167  crossref
    3. Brociek R., Pleszczynski M., Witula R., Lorenc P., “Transformations Preserving Nonsingularity, Trace and Spectrum of Matrices”, Application of Mathematics in Technical and Natural Sciences, AIP Conference Proceedings, 1629, ed. Todorov M., Amer Inst Physics, 2014, 344–348  crossref  adsnasa  isi  scopus
    4. Kh. D. Ikramov, “Effective Algorithms for Decomplexifying a Matrix by Unitary Similarities or Congruences”, Math. Notes, 92:6 (2012), 767–772  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. A. K. Abdikalykov, Kh. D. Ikramov, “Simultaneous decomplexification of a pair of complex matrices via a unitary similarity transformation”, J. Math. Sci. (N. Y.), 182:6 (2012), 745–747  mathnet  crossref  mathscinet
    6. Kh. D. Ikramov, “How to distinguish between the latently real matrices and the block quaternions?”, J. Math. Sci. (N. Y.), 182:6 (2012), 779–781  mathnet  crossref  mathscinet
    7. Kh. D. Ikramov, “On latently real matrices and block quaternions”, J. Math. Sci. (N. Y.), 176:1 (2011), 25–28  mathnet  crossref
    8. Kh. D. Ikramov, “Unitary similarity of entrywise conjugate matrices and unitary decomplexification”, J. Math. Sci. (N. Y.), 176:1 (2011), 29–31  mathnet  crossref
    9. Ikramov Kh.D., “Constructive sufficient conditions for the existence of a unitary similarity transformation that converts a given complex matrix into a real one”, Dokl. Math., 82:1 (2010), 563–565  crossref  mathscinet  zmath  isi  elib  scopus
    Citing articles in Google Scholar: Russian citations, English citations
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