  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 [Hartfiel, D. J. Dense sets of diagonalizable matrices. Proc.
Amer. Math. Soc., 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. We are interested
in finding a few 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 of our results are generalizing the results of
Hartfiel. Also, we study the existence of dense subsets of matrices with
distinct singular values, distinct analytic eigenvalues, and distinct analytic
singular values, respectively, in the subsets of the set of all real or complex
matrices.
