Algorithms for multidimensional spectral factorization and sum of squares
type: Item
instance of: scholarly article
title: Algorithms for multidimensional spectral factorization and sum of squares
zbMATH Open document ID: 1155.65032
author: Diego Napp
author: Harry Trentelman
published in: Linear Algebra and its Applications
publication date: 2008-08-06T00:00:00Z
full work available at URL: 2008LinAlgApplNappAvelli.pdf
review text: Algorithms for multidimensional spectral factorization and sum of squares of polynomial matrices are developed. In the problem of multidimensional spectral factorization for a real \(n\)-variable polynomial matrix \(Z(\xi)\) with the property \(Z^T(-\xi) = Z(\xi)\) another real \(n\)-variable polynomial matrix \(F(\xi)\), the so-called spectral factor, has to be computed such that \(Z(\xi) = F^T(-\xi)F(\xi)\), where \(\xi = (\xi_1,\ldots ,\xi_n)\) denotes an \(n\)-dimensional indeterminate. The problem of sum of squares is the following: For a real \(n\)-variable polynomial matrix \(Z(\xi)\) with \(Z^T(\xi) = Z(\xi)\), \(Z(\xi) \geq 0\) for all \(\xi \in R^n\) a real \(n\)-variable polynomial matrix \(F(\xi)\) has to be determined such that \(Z(\xi) = F^T(\xi)F(\xi)\). It is shown how the problem of multidimensional spectral factorization can be reduced to the factorization of a real symmetric constant matrix. This is achieved by associating a \(2n\)-variable polynomial matrix to the matrix \(Z(\xi)\). Two algorithms for the multidimensional spectral factorization are presented. In the first one, one has to solve a linear matrix inequality and in the second one a linear eigenvalue problem. In the second case also spectral factors which are rational matrices are considered. In the last section the problem of sum of squares is connected with a \(2n\)-variable polynomial matrix and a linear matrix inequality.
reviewed by: Michael Jung
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 5307054
zbMATH Keywords: polynomial multidimensional spectral factorization
zbMATH Keywords: two-variable polynomial matrices
zbMATH Keywords: quadratic differential forms
zbMATH Keywords: dissipativity
zbMATH Keywords: sum of squares
zbMATH Keywords: linear matrix inequality
zbMATH Keywords: algorithms
zbMATH Keywords: linear eigenvalue problem
MaRDI profile type: Publication
OpenAlex ID: W2068495657
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DOI: 10.1016/J.LAA.2007.06.003
description: scientific article; zbMATH DE number 5307054