An Effective Algorithm for Computing the Numerical Range

In this paper, we collect a few fairly well known facts about the numerical range and assemble them into an effective algorithm for computing the numerical range of an n× n matrix. Particular attention is paid to the case in which a line segment is embedded in the boundary of the numerical range, a case in which multiplicity is present. The result is a parametrization of the curve that forms the boundary of the numerical range. A Matlab implementation of the algorithm is included. If A is an n× n complex matrix, the numerical range of A is the subset of the complex plane given by w(A) = {〈Av, v〉 : v ∈ C with ‖v‖ = 1} The classical Toeplitz–Hausdorff Theorem [2] asserts that the numerical range of every n × n matrix is a convex set. One approach to the proof of this theorem is to show by direct computation that the numerical range of every 2×2 matrix is an ellipse with its interior and observe that if 〈Au, u〉 and 〈Av, v〉 are two points of the numerical range of A, then the numerical range of the compression of A to the subspace spanned by u and v is an ellipse that contains these two points and is contained in the numerical range of A. Thus, since the ellipse is a convex set, the line segment joining 〈Au, u〉 ∗Supported in part by National Science Foundation Grants DMS 92-06965 and DMS 95-00870.