On separation of eigenvalues by certain matrix subgroups

On the separation of eigenvalues by the permutation group

“…Therefore, we can suppose that for all X ∈ H the products XA are nilpotent. Since tr (XA) = α + tr (X A ) = , the trace of X A is constant for all X ∈ H. Lemma 4.3 from [2] implies that α = and A = .…”

“…Is it true that for every × invertible complex matrix A there is a permutation matrix P ∈ S such that the product PA has distinct eigenvalues? That is, is every × invertible complex matrix S -separable (see [2]). …”

“…Since C is algebraically closed, H is irreducible if and only if it possesses a basis of C n×n . In [2] it is shown that for an irreducible group H every non-zero matrix is H-semiseparable. For an excellent treatment of irreducible matrix subgroups see [7].…”

“…By Mon denote the group of all invertible n × n monomial matrices and by Uni the group of all n × n unitary matrices. In [2] it is shown that every n × n complex matrix is Mon and Uni-separable.…”