Companion matrices: Reducibility, numerical ranges and similarity to contractions

In this paper, we study some unitary-equivalence properties of the companion matrices. We obtain a criterion for a companion matrix to be reducible and show that the numerical range of a companion matrix is a circular disc centered at the origin if and only if the matrix equals the (nilpotent) Jordan block. However, the more general assertion that a companion matrix is determined by its numerical range turns out to be false. We also determine, for an n×n matrix A with eigenvalues in the open unit disc, the defect index of a contraction to which A is similar.