Singular points of the ternary polynomials associated with 4-by-4 matrices

Chien, Mao-Ting; Nakazato, Hiroshi

doi:10.13001/1081-3810.1554

Cited by 26 publications

(27 citation statements)

References 11 publications

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“…In particular, we show that a certain type of singularity is sufficient, but not necessary, to imply k(A) > 2. In Section 5, we consider unitarily irreducible matrices A with reducible base polynomials, extending Chien and Nakazato's characterization in [3]. Throughout the paper, we give examples of matrices with different singularities and values of k(A).…”

Section: Definition 2 Letmentioning

confidence: 99%

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Singularities of base polynomials and Gau–Wu numbers

Camenga

Deaett

Rault

et al. 2019

Linear Algebra and its Applications

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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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Section: Definition 2 Letmentioning

confidence: 99%

“…Inspired by Chien and Nakazato [3], we use the following tools to study the numerical range. Let A ∈ M n (C) and let where A * is the conjugate transpose of A.…”

Section: Introductionmentioning

confidence: 99%

Singularities of base polynomials and Gau–Wu numbers

Camenga

Deaett

Rault

et al. 2019

Linear Algebra and its Applications

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“…The classification of 4 × 4 matrices [10] shows that irreducible elliptic curves F(t, x, y), and assume that the singular points of the curve F(t, x, y) = 0 are the above three types. We divide the type I I I j into two subtypes.…”

Section: Quartic Hyperbolic Formsmentioning

confidence: 99%

“…We refer the reader to [10,11] for the genus of a general irreducible algebraic curve. An irreducible curve C F is rational if g(C F ) = 0, and elliptic if g(C F ) = 1.…”

Section: Introductionmentioning

confidence: 99%

Elliptic modular invariants and numerical ranges

Chien

Nakazato

2014

Linear and Multilinear Algebra

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“…Concrete tests to determine the shape from the entries of the 3 × 3 matrix are in [9]. Work has also been done to generalize Kippenhahn's result to 4 × 4 matrices [2]. More generally, [8] includes a necessary and sufficient condition for a convex subset of C to be the numerical range of some n × n matrix; the result follows from work on linear matrix inequalities.…”

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confidence: 99%

Trace conditions for symmetry of the numerical range

Deaett¹,

Lafuente-Rodriguez²,

Marín³

et al. 2013

ELA

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Abstract. A subset S of the complex plane has n-fold symmetry about the origin (n-sato) if z ∈ S implies e 2π n z ∈ S. The 3 × 3 matrices A for which the numerical range W (A) has 3-sato have been characterized in two ways. First, W (A) has 3-sato if and only if the spectrum of A has 3-sato while tr(A 2 A * ) = 0. In addition, W (A) has 3-sato if and only if A is unitarily similar to an element of a certain family of generalized permutation matrices. Here it is shown that for an n × n matrix A, if a specific finite collection of traces of words in A and A * are all zero, then W (A) has n-sato. Further, this condition is shown to be necessary when n = 4. Meanwhile, an example is provided to show that the condition of being unitarily similar to a generalized permutation matrix does not extend in an obvious way.

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Singular points of the ternary polynomials associated with 4-by-4 matrices

Cited by 26 publications

References 11 publications

Singularities of base polynomials and Gau–Wu numbers

Singularities of base polynomials and Gau–Wu numbers

Elliptic modular invariants and numerical ranges

Trace conditions for symmetry of the numerical range

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