Determinantal representations of hyperbolic forms via weighted shift matrices

Chien, Mao-Ting; Nakazato, Hiroshi

doi:10.1016/j.amc.2015.02.019

Cited by 8 publications

(5 citation statements)

References 22 publications

(23 reference statements)

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“…Chien and Nakazato posed the converse problem and were interested in its connection to numerical ranges [3]. In view of [13], we prove that a hyperbolic plane curve, invariant under the action of the cyclic group C n , has a determinantal representation admitted by some cyclic weighted shift matrix with complex entries.…”

Section: Introductionmentioning

confidence: 99%

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Determinantal representations of invariant hyperbolic plane curves

Lentzos

Pasley

2018

Linear Algebra and its Applications

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We study hyperbolic polynomials with nice symmetry and express them as the determinant of a Hermitian matrix with special structure. The goal of this paper is to answer a question posed by Chien and Nakazato in 2015. By properly modifying a determinantal representation construction of Dixon (1902), we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix. Moreover, if the polynomial is invariant under the action of the dihedral group of order n, the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries. arXiv:1707.07724v1 [math.AG]

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Section: Introductionmentioning

confidence: 99%

“…The example in Section 3 of [3] shows there exists matrix A so f A ∈ C[t, x, y] Cn n and A is not unitarily equivalent to any cyclic weighted shift matrix (with positive weights). Theorem 1.6 proves there must exist some cyclic weighted shift matrix with the same numerical range of A, even if they are not unitarily equivalent.…”

Section: Introductionmentioning

confidence: 99%

Determinantal representations of invariant hyperbolic plane curves

Lentzos

Pasley

2018

Linear Algebra and its Applications

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“…Typical hyperbolic ternary forms may admit determinantal representation by special matrices. For instance, it is proved in [11] that hyperbolic ternary forms satisfying weak symmetry admit determinantal representations via cyclic weighted shift matrices for lower degrees. Lentzos and Pasley [12] solved the problem for general degrees.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices †

Chien

Nakazato

2018

Symmetry

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Let A be an n × n complex matrix. Assume the determinantal curve V A = {[(x, y, z)] ∈ CP 2 : F A (x, y, z) = det(x (A) + y (A) + zI n) = 0} is a rational curve. The Fiedler formula provides a complex symmetric matrix S satisfying F S (x, y, z) = F A (x, y, z). It is also known that every Toeplitz matrix is unitarily similar to a symmetric matrix. In this paper, we investigate the unitary similarity of the symmetric matrix S and the matrix A in the Fiedler theorem for a specific parametrized family of 4 × 4 nilpotent Toeplitz matrices A. We show that there are either one or at least three unitarily inequivalent symmetric matrices which admit the determinantal representation of the ternary from F A (x, y, z) associated to the specific 4 × 4 nilpotent Toeplitz matrices.

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“…In [1], the classification of the curve F T (x, y, z) = 0 is applied to categorize the shapes of the numerical ranges of 4 × 4 matrices. In [2] and [3], we discussed rational curves and elliptic curves in the context of numerical ranges. For any pair (x 0 , y 0 ) of real numbers, the algebraic equation F T (x 0 , y 0 , z) = 0 has n real solutions in z because x 0 (T )+y 0 (T ) is a Hermitian matrix.…”

Section: Introductionmentioning

confidence: 99%

Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials

Chien¹,

Nakazato²

2017

Ann. Funct. Anal.

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The numerical range of a matrix, according to Kippenhahn, is determined by a hyperbolic determinantal form of linear Hermitian matrices associated to the matrix. On the other hand, using Riemann theta functions, Helton and Vinnikov confirmed that a hyperbolic form always admits a determinantal representation of linear real symmetric matrices. The Riemann matrix of the hyperbolic curve plays the main role in the existence of real symmetric matrices. In this article, we implement computations of the Riemann matrix and the Abel-Jacobi variety of the hyperbolic curve associated to a determinantal polynomial of a matrix. Further, we prove that the lattice of the Abel-Jacobi variety is decomposed into the direct sum of two orthogonal lattices for some 4 × 4 Toeplitz matrices.W (T ) = {ξ * T ξ : ξ ∈ C n , ξ * ξ = 1}.

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Determinantal representations of hyperbolic forms via weighted shift matrices

Cited by 8 publications

References 22 publications

Determinantal representations of invariant hyperbolic plane curves

Determinantal representations of invariant hyperbolic plane curves

Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices †

Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials

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