Normal matrices with a dominant eigenvalue and an eigenvector with no zero entries

Abstract

We say that a square complex matrix is dominant if it has an algebraically simple eigenvalue whose modulus is strictly greater than the modulus of any other eigenvalue; such an eigenvalue and any associated eigenvector are also said to be dominant. We explore inequalities that are sufficient to ensure that a normal matrix is dominant and has a dominant eigenvector with no zero entries. For a real symmetric matrix, these inequalities force the entries of a dominant real eigenvector to have a prescribed sign pattern. In the cases of equality in our inequalities, we find that exceptional extremal matrices must have a very special form.