In this note we are, essentially, concerned with generalizations of the (known) fact tha t an n • matr ix with n linearly independent eigenvectors all corresponding to real eigenvalues is similar to a hermitian matrix, and can consequently be transformed into its conjugate transpose by a positive definite hermitian similarity. We first establish, fo r ' any positive integer m, an analogous necessary and sufficient condition tha t a given square complex matr ix A should have a set of real eigenvalues, not necessarily all distinct, to which therecorrespond at least m linearly,independent eigenvectors; this of course implies a corresponding result about pure imaginary eigenvalues. We also obtain an analogous result concerning eigenvalues of modulus unity: As a simple application of our more general results, we establish, in Theorem 4, the reality of the eigenvalues of a certain rather special type of matrix. Throughout, we shall use A = (aij) to denote an arbi t rary n • n complex matrix, and A* will denote the transposed conjugate matr ix; we denote the rank of A by r(A).