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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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.

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Dense sets of diagonalizable matrices
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by D. J. Hartfiel PDF
Proc. Amer. Math. Soc. 123 (1995), 1669-1672 Request permission

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 \[ W = \left \{ {A:A = \left [ {\begin {array}{*{20}{c}} 0 \hfill & 0 \hfill \\ a \hfill & 0 \hfill \\ \end {array} } \right ]\quad {\text {where}}\;a \in R} \right \}\] 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
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1669-1672
  • MSC: Primary 15A18
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1264813-7
  • MathSciNet review: 1264813