Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for γ = 0.05 and far-field loading σ = 0.01, 0.02, 0.04. 47 3.3 Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for far-field loading σ 0 = 0.02 andγ = 0.

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In 2013, Gau and Wu introduced a unitary invariant, denoted by k(A), of an n × n matrix A, which counts the maximal number of orthonormal vectors x j such that the scalar products Ax j , x j lie on the boundary of the numerical range W (A). We refer to k(A) as the Gau-Wu number of the matrix A. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify k(A). This continues the work of Wang and Wu [14] classifying the Gau-Wu numbers for 3 × 3 matrices. Our focus on singularities is inspired by Chien and Nakazato [3], who classified W (A) for 4 × 4 unitarily irreducible A with irreducible base curve according to singularities of that curve. When A is a unitarily irreducible n × n matrix, we give necessary conditions for k(A) = 2, characterize k(A) = n, and apply these results to the case of unitarily irreducible 4 × 4 matrices. However, we show that knowledge of the singularities is not sufficient to determine k(A) by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different k(A). In addition, we extend Chien and Nakazato's classification to consider unitarily irreducible A with reducible base curve and show that we can find corresponding matrices with identical base curve but different k(A). Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky [5], generalizing their result in the process.Re A = (A + A * ) /2 and Im A = (A − A * ) /2i, 1991 Mathematics Subject Classification. Primary 15A60.

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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.