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

 

Dense sets of diagonalizable matrices


Author: D. J. Hartfiel
Journal: Proc. Amer. Math. Soc. 123 (1995), 1669-1672
MSC: Primary 15A18
MathSciNet review: 1264813
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Abstract: This paper provides necessary and sufficient conditions for a subspace of matrices to contain a dense set of matrices having distinct eigenvalues.

A well-known and useful result in linear algebra is that matrices with distinct eigenvalues are dense in the set of $ n \times n$ matrices. This result, however, does not hold for subspaces of matrices in general. For example, the subspace

$\displaystyle W = \left\{ {A:A = \left[ {\begin{array}{*{20}{c}} 0 \hfill & 0 \... ...fill & 0 \hfill \\ \end{array} } \right]\quad {\text{where}}\;a \in R} \right\}$

contains no matrix with distinct eigenvalues. In this paper we give necessary and sufficient conditions for a subspace of matrices to contain a dense set of matrices having distinct eigenvalues. The result is then applied to subspaces of matrices determined by specified 0 patterns.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1264813-7
PII: S 0002-9939(1995)1264813-7
Article copyright: © Copyright 1995 American Mathematical Society