Crawford numbers of companion matrices

The (generalized) Crawford number C(A) of an n-by-n complex matrix A is, by definition, the distance from the origin to the boundary of the numerical range W(A) of A. If A is a companion matrix (Formula Presented) then it is easily seen that C(A) ≥ cos(π/n). The main purpose of this paper is to determine when the equality C(A) = cos(π/n) holds. A sufficient condition for this is that the boundary of W(A) contains a point λ for which the subspace of ℂⁿ spanned by the vectors x with 〈Ax,x〉 = λ||x||² has dimension 2, while a necessary condition is (Formula Presented) for some real θ. Examples are given showing that in general these conditions are not simultaneously necessary and sufficient. We then prove that they are if A is (unitarily) reducible. We also establish a lower bound for the numerical radius w(A) of A: w(A) ≥ cos(π/(n+1)), and show that the equality holds if and only if A is equal to the n-by-n Jordan block.