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Determinantal representation of trigonometric polynomial curves via Sylvester method. (English) Zbl 1283.15025

Let, for an \(n \times n\) complex matrix \(A\), \(\; F_A(t,x,y) := \det(t \, I_n+x \, \mathfrak{R}(A)+y \, \mathfrak{I}(A))\) denote the real ternary form associated with the matrix \(A\), and let \[ \phi(\theta) = \sum_{k=-n}^n c_k \exp (ik\theta) \] be a trigonometric polynomial. Using Sylvester matrices, these authors present an algorithm to construct \(2n \times 2n\) matrices \(C_1\), \(C_2\), \(C_3\) such that, for a given trigonometric polynomial \(\phi(\theta),\) \[ \det (C_1 + \mathfrak{R}(\phi(\theta)) C_2 + \mathfrak{I} (\phi(\theta)) C_3) = 0. \] If \(\phi\) has the special form \(\phi(\theta) = \exp(in\theta) + a \exp(-im\theta)\), \(0 \leq \theta \leq 2\pi\), \(0 < m < n\), \(0 < a < 1\), then the matrices \(C_1\), \(C_2\), \(C_3\) are Hermitian, \(C_1\) is positive definite, and the relation \[ F_{C_0}(t,x,y) \det(C_1) = \det(tC_1+xC_2+yC_3), \] holds, where \(C_0 = C_1^{-1/2} (C_2+iC_3) C_1^{-1/2}.\)

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A12 Numerical range, numerical radius
42A05 Trigonometric polynomials, inequalities, extremal problems
15B57 Hermitian, skew-Hermitian, and related matrices
15B05 Toeplitz, Cauchy, and related matrices
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References:

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