Singularities of base polynomials and Gau–Wu numbers

Abstract

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.

On the Geometry of Numerical Ranges Over Finite Fields

“…In this section we will prove Proposition 3, after developing the necessary tools. We begin by building on a result of [3] to establish a correspondence between eigenvalues of the Hermitian part of a matrix and the multiplicity of a singularity.…”

“…For a matrix A ∈ M n (F q 2 ), the numerical range of A, as defined in [5], is the set W (A) = Av, v : v, v = 1, v ∈ F n q 2 . The classical complex number numerical range was first studied by Kippenhahn [8,10] and has been completely classified for n × n matrices up to dimension n = 4 [7,4,3]. The most recent of these classifications relies on the boundary generating curve of the matrix [4,3].…”

“…The classical complex number numerical range was first studied by Kippenhahn [8,10] and has been completely classified for n × n matrices up to dimension n = 4 [7,4,3]. The most recent of these classifications relies on the boundary generating curve of the matrix [4,3]. We will now define this curve in detail, as it is pivotal to the results in this paper.…”

Kippenhahn curves of some tridiagonal matrices

“…The only difference in the statement of Theorem 3 from the material already contained in [3] is the explicit formula (2.5) for the s-numbers σ j of the n/2 × n/2 matrix X with the entries…”

“…Visualizing reciprocal 4-by-4 matrices as points {A 1 , A 2 , A 3 } in R 3 + we see that those with elliptical numerical ranges form a 2-dimensional manifold M 4 described by (3.3). These equations show that M 4 contains the ray (2.4), as it should according to Theorem 3.…”