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On Matrix Subspaces with Trivial Quadratic Kernels

  • Alexey Tretyakov
  • Eugene Tyrtyshnikov
  • Alexey Ustimenko
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 30)

Abstract

Some subspaces of real matrices of the same order may contain nonsingular matrices, some may not. We prove that if the maximal rank matrix in the given subspace with trivial quad ratic kernel is symmetric, then it must be nonsingular. It immediately follows that any subspace of symmetric matrices with trivial quad ratic kernel contains a nonsingular matrix. We present some particular cases when this holds true without the assumption about symmetry. Whether this remains valid in the general case of real nonsymmetric matrices we still do not know.

Keywords

Matrix subspaces Quadratic kernels 

Notes

Acknowledgements

This work is supported by the Program of the Presidium of the Russian Academy of Sciences no. 01 “Fundamental Mathematics and its Applications” under grant PRAS-18-01.

References

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    Tretyakov, A., Tyrtyshnikov, E., Ustimenko, A.: The triviality condition for kernels of quad ratic mappings and its application in optimization methods. Russ. J. Numer. Anal. Math. Model. 32(4), 1–9 (2017)Google Scholar
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    Tyrtyshnikov, E.E.: Foundations of Algebra. FIZMATLIT, Moscow (2017, in Russian)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexey Tretyakov
    • 1
    • 2
  • Eugene Tyrtyshnikov
    • 1
    • 3
    • 4
  • Alexey Ustimenko
    • 3
  1. 1.Faculty of SciencesSiedlce UniversitySiedlcePoland
  2. 2.System Research InstitutePolish Academy of SciencesWarsawPoland
  3. 3.Marchuk Institute of Numerical Mathematics of Russian Academy of SciencesMoscowRussia
  4. 4.Lomonosov Moscow State UniversityMoscowRussia

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