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Polynomial matrices are widely studied in … Polynomial matrices are widely studied in the fields of systems theory and control theory and have seen other uses relating to stable polynomials. In stability theory, Spectral Factorization has been used to find determinantal matrix representations for bivariate stable polynomials and real zero polynomials. A key tool used to study these is a matrix factorization known as either the Polynomial Matrix Spectral Factorization or the Matrix Fejer–Riesz Theorem. Given a univariate positive polynomial , a polynomial which takes on non-negative values for any real input , the Fejer–Riesz Theorem yields the polynomial spectral factorization . Results of this form are generically referred to as Positivstellensatz. Considering positive definiteness as the matrix analogue of positivity, Polynomial Matrix Spectral Factorization provides a similar factorization for polynomial matrices which have positive definite range. This decomposition also relates to the Cholesky decomposition for scalar matrices . This result was originally proven by Wiener in a more general context which was concerned with integrable matrix-valued functions that also had integrable log determinant. Because applications are often concerned with the polynomial restriction, simpler proofs and individual analysis exist focusing on this case. Weaker positivstellensatz conditions have been studied, specifically considering when the polynomial matrix has positive definite image on semi-algebraic subsets of the reals. Many publications recently have focused on streamlining proofs for these related results. This article roughly follows the recent proof method of Lasha Ephremidze which relies only on elementary linear algebra and complex analysis. Spectral Factorization is used extensively in linear–quadratic–Gaussian control. Because of this application there have been many algorithms to calculate spectral factors. Some modern algorithms focus on the more general setting originally studied by Wiener. In the case the problem is known as polynomial spectral factorization, or Fejer-Riesz Theorem, and has many classical algorithms. Some modern algorithms have used Toeplitz matrix advances to speed up factor calculations. advances to speed up factor calculations.
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| rdfs:comment |
Polynomial matrices are widely studied in … Polynomial matrices are widely studied in the fields of systems theory and control theory and have seen other uses relating to stable polynomials. In stability theory, Spectral Factorization has been used to find determinantal matrix representations for bivariate stable polynomials and real zero polynomials. A key tool used to study these is a matrix factorization known as either the Polynomial Matrix Spectral Factorization or the Matrix Fejer–Riesz Theorem.ization or the Matrix Fejer–Riesz Theorem.
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Polynomial matrix spectral factorization
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