On separation of eigenvalues by certain matrix subgroups

Let A be an n×n complex matrix with rank r. It is shown that there are a monomial matrix M and a unitary matrix U such that each of the matrices MA and UA has r distinct non-zero eigenvalues. If H is an irreducible subgroup of GLn(C) and A≠0, it is shown that there is an X∈H such that XA has at least two distinct eigenvalues.