3-by-3 matrices with elliptical numerical range revisited

Abstract

Abstract. According to Kippenhahn's classification, numerical ranges W (A) of unitarily irreducible 3 × 3 matrices A come in three possible shapes, an elliptical disk being one of them. The known criterion for the ellipticity of W (A) consists of several equations, involving the eigenvalues of A. It is shown herein that the set of 3 × 3 matrices satisfying these conditions is nowhere dense, i.e., one of the necessary conditions can be violated by an arbitrarily small perturbation of the matrix, and therefore by an insufficiently good numerical approximation of the eigenvalues.Moreover, necessary and sufficient conditions for a real A to have an elliptical W (A) are derived, involving only the matrix coefficients and not requiring the knowledge of the eigenvalues. A particular case of real companion matrices is considered in detail.

Classification of joint numerical ranges of three hermitian matrices of size three

“…If F is unitarily irreducible, then W (F ) is an ellipse, the convex hull of a quartic curve (with a flat portion on the boundary), or the convex hull of a sextic curve (an oval). Kippenhahn's result for 3-by-3 matrices was expressed in terms of matrix invariants and matrix entries of F 1 + i F 2 , see [35,49,47,54]. The boundary generating curve was also used [16] to find a classification of the numerical range of a 4-by-4 matrix.…”

Pre-images of extreme points of the numerical range, and applications

“…This implies further that the set of 3-by-3 matrices who have an elliptical numerical range is closed relative to the set of non-normal matrices. The fact that E 3 is nowhere dense was derived from the same criterion in [26], see Prop. 3.1 there.…”

“…This implies further that the set of 3-by-3 matrices who have an elliptical numerical range is closed relative to the set of non-normal matrices. The fact that E 3 is nowhere dense was derived from the same criterion in [26], see Prop. The statements ( 2) and (3) can be refined using Kippenhahn's classification in Remarks 4.1 and 4.2 and the property that the numerical ranges of a converging sequence of matrices converge to the numerical range of the limit matrix (Rem.…”

“…Finally, in Sec. 4 we combine observations from [14,22,21,28,23,26] with Thm. 1.1 to compute closures of subsets of 3-by-3 matrices having the same shape of the numerical range.…”