Dense sets of diagonalizable matrices

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.