On the dense subsets of matrices with distinct eigenvalues and distinct singular values

Abstract

It is well known that the set of all $n \times n$ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $n \times n$ matrices. In [D.J. Hartfiel. Dense sets of diagonalizable matrices. {\em Proceedings of the American Mathematical Society}, 123(6):1669--1672, 1995.], the author established a necessary and sufficient condition for a subspace of the set of all $n \times n$ matrices to have a dense subset of matrices with distinct eigenvalues. The objective of this article is to identify necessary and sufficient conditions for a subset of the set of all $n \times n$ real or complex matrices to have a dense subset of matrices with distinct eigenvalues. Some results of Hartfiel are extended, and the existence of dense subsets of matrices with distinct singular values in the subsets of the set of all real or complex matrices is studied. Furthermore, for a matrix function $F(x)$, defined on a closed and bounded interval whose entries are analytic functions, it is proved that the set of all points for which the matrix $F(x)$ has repeated analytic eigenvalues/analytic singular values is either a finite set or the whole domain of the function $F$.