Numerical range of a matrix: some effective criteria

Abstract

Algorithms are presented which decide, for a given complex number w and a given complex n× n matrix S, whether w is in the numerical range W( S) of S, whether w is a boundary point of W( S), whether w is an extreme point of W( S), whether w is a bare point of W( S), and whether w is a vertex of W( S). Further algorithms decide whether W( S) intersects a given line (or a given ray), whether W( S) is included in a given open half plane (or a given closed half plane), and, for a given real number r, whether the numerical radius ρ s of S is > r, whether ρ s = r, and whether ρ s ≥ r. A simple effective criterion for H-stability is also given: a nonsingular H-semistable matrix S is H-stable iff the nullity of ( S+ S ∗) S -1( S+ S ∗) is twice the nullity of S+ S ∗. The computations involved in all these algorithms are elementary (rational operations, the max operation on pairs of real numbers, the degree of a nonzero polynomial, and the number of sign variations in the coefficients of a nonzero real polynomial), must be carried out exactly, and give exact (i.e., 100% reliable) results. Examples are worked out to illustrate the application of some of the algorithms.