Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1995-1264813-7
MathSciNet review: 1264813
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Joel Franklin, Matrix theory, Prentice-Hall, Englewood Cliffs, NJ, 1968. MR 0237517 (38:5798)
  • [2] D. J. Hartfiel, A simplified form for nearly reducible and nearly decomposable matrices, Proc. Amer. Math. Soc. 24 (1970), 388-393. MR 0252415 (40:5635)
  • [3] -, Tracking in matrix systems, Linear Algebra Appl. 16 (1992), 233-250. MR 1149757 (92k:15046)
  • [4] Nathan Jacobson, Basic algebra I, W. H. Freeman, San Francisco, CA, 1974. MR 0356989 (50:9457)
  • [5] Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, MA, 1964. MR 0162808 (29:112)
  • [6] Herbert Simon and Albert Ando, Aggregation of variables in dynamic systems, Econometrica 29 (1961), 111-138.
  • [7] Richard Sinkhorn, Concerning a conjecture of Marshall Hall, Proc. Amer. Math. Soc. 21 (1969), 197-201. MR 0241440 (39:2780)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A18

Retrieve articles in all journals with MSC: 15A18


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1264813-7
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society