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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On linear transformations preserving at least one eigenvalue
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by S. Akbari and M. Aryapoor
Proc. Amer. Math. Soc. 132 (2004), 1621-1625
DOI: https://doi.org/10.1090/S0002-9939-03-07262-9
Published electronically: December 5, 2003
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Abstract:

Let $F$ be an algebraically closed field and $T: M_n(F) \longrightarrow M_n(F)$ be a linear transformation. In this paper we show that if $T$ preserves at least one eigenvalue of each matrix, then $T$ preserves all eigenvalues of each matrix. Moreover, for any infinite field $F$ (not necessarily algebraically closed) we prove that if $T: M_n(F) \longrightarrow M_n(F)$ is a linear transformation and for any $A\in M_n(F)$ with at least an eigenvalue in $F$, $A$ and $T(A)$ have at least one common eigenvalue in $F$, then $T$ preserves the characteristic polynomial.
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Bibliographic Information
  • S. Akbari
  • Affiliation: Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11365-9415, Tehran, Iran
  • Email: s_akbari@sina.sharif.ac.ir
  • M. Aryapoor
  • Affiliation: Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11365-9415, Tehran, Iran
  • Email: aryapoor2002@yahoo.com
  • Received by editor(s): December 17, 2002
  • Received by editor(s) in revised form: February 27, 2003
  • Published electronically: December 5, 2003
  • Communicated by: Joseph A. Ball
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 1621-1625
  • MSC (2000): Primary 15A04, 47B49
  • DOI: https://doi.org/10.1090/S0002-9939-03-07262-9
  • MathSciNet review: 2051122