Skip to Main Content

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 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On linear transformations preserving at least one eigenvalue
HTML articles powered by AMS MathViewer

by S. Akbari and M. Aryapoor PDF
Proc. Amer. Math. Soc. 132 (2004), 1621-1625 Request permission

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.
References
  • William C. Brown, Matrices over commutative rings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 169, Marcel Dekker, Inc., New York, 1993. MR 1200234
  • Frobenius, G., Uber die Darstellung der endlichen Gruppen durch linear Substitutionen, Sitzungsber Deutsch. Akad. Wiss. Berlin, 1897, 994-1015.
  • Marvin Marcus and B. N. Moyls, Transformations on tensor product spaces, Pacific J. Math. 9 (1959), 1215–1221. MR 108503
  • Marvin Marcus and Roger Purves, Linear transformations on algebras of matrices: the invariance of the elementary symmetric functions, Canadian J. Math. 11 (1959), 383–396. MR 105425, DOI 10.4153/CJM-1959-039-4
  • Pierce, S., et al., A Survey of Linear Preserver Problems, Linear and Multilinear Algebra 33 (1992), 1-129.
  • Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR 674652
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 15A04, 47B49
  • Retrieve articles in all journals with MSC (2000): 15A04, 47B49
Additional 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