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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On linear transformations preserving at least one eigenvalue


Authors: S. Akbari and M. Aryapoor
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1621-1625
MSC (2000): Primary 15A04, 47B49
Published electronically: December 5, 2003
MathSciNet review: 2051122
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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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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

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07262-9
PII: S 0002-9939(03)07262-9
Keywords: Linear transformation, preserve, eigenvalue
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
Article copyright: © Copyright 2003 American Mathematical Society