Numerical ranges and Geršgorin discs

For a complex matrix , let W(A) be its numerical range, and let G(A) be the convex hull of and .It is known that W(A) is always contained in G(A) and hence in G′(A). In this paper, we consider conditions for W(A) to be equal to G(A) or G′(A). We show that if W(A) = G′(A), then the boundary of W(A) consists only of circular arcs and line segments. If, moreover, A is unitarily irreducible, then W(A) is a circular disc. (Almost) complete characterizations of 2-by-2 and 3-by-3 matrices A for which W(A) = G′(A) are obtained. We also give criteria for the equality of W(A) and G(A). In particular, such A’s among the permutationally irreducible ones must have even sizes. We also characterize those A’s with size 2 or 4 which satisfy W(A) = G(A).