CRAWFORD NUMBERS OF COMPANION MATRICES

Abstract

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 [GRAPHICS] then it is easily seen that C(A) >= cos(pi/n). The main purpose of this paper is to determine when the equality C(A) = cos(pi/n) holds. A sufficient condition for this is that the boundary of W(A) contains a point lambda for which the subspace of C-n spanned by the vectors x with < Ax, x > = lambda parallel to x parallel to(2) has dimension 2, while a necessary condition is Sigma(n-2)(j=0) a(n-j)e((n-j)i theta) sin ((j + 1)pi/n) = sin(pi/n) for some real theta. 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(pi/(n+ 1)), and show that the equality holds if and only if A is equal to the n-by-n Jordan block.