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Computing matrix symmetrizers. Part 2: new methods using eigendata and linear means; a comparison

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Interdisciplinar de Sistemas Complejos (GISC)es
dc.contributor.authorMartínez Dopico, Froilán César
dc.contributor.authorUhlig, Frank
dc.date.accessioned2015-12-02T12:31:42Z
dc.date.available2017-07-10T22:00:09Z
dc.date.issued2015-07-10
dc.description.abstractOver any field F every square matrix A can be factored into the product of two symmetric matrices as A = S1 . S2 with S_i = S_i^T ∈ F^(n,n) and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910. Frobenius’ symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such that SA is symmetric, was inspired by an iterative linear systems algorithm of Huang and Nong (2010) in 2013 [29, 30]. The resulting iterative algorithm has solved this computational problem over R and C, but at high computational cost. This paper develops and tests another linear equations solver, as well as eigen- and principal vector or Schur Normal Form based algorithms for solving the matrix symmetrizer problem numerically. Four new eigendata based algorithms use, respectively, SVD based principal vector chain constructions, Gram-Schmidt orthogonalization techniques, the Arnoldi method, or the Schur Normal Form of A in their formulations. They are helped by Datta’s 1973 method that symmetrizes unreduced Hessenberg matrices directly. The eigendata based methods work well and quickly for generic matrices A and create well conditioned matrix symmetrizers through eigenvector dyad accumulation. But all of the eigen based methods have differing deficiencies with matrices A that have ill-conditioned or complicated eigen structures with nontrivial Jordan normal forms. Our symmetrizer studies for matrices with ill-conditioned eigensystems lead to two open problems of matrix optimization.en
dc.description.sponsorshipThis research was partially supported by the Ministerio de Economía y Competitividad of Spain through the research grant MTM2012-32542.en
dc.description.statusPublicado
dc.format.extent33
dc.format.mimetypeapplication/pdf
dc.identifier.bibliographicCitationLinear Algebra and its Applications (2015) July
dc.identifier.doi10.1016/j.laa.2015.06.031
dc.identifier.issn0024-3795
dc.identifier.publicationfirstpage1
dc.identifier.publicationissueJulyen
dc.identifier.publicationlastpage33
dc.identifier.publicationtitleLinear algebra and its applicationsen
dc.identifier.urihttps://hdl.handle.net/10016/22064
dc.identifier.uxxiAR/0000017410
dc.language.isoeng
dc.publisherElsevieren
dc.relation.projectIDGobierno de España. MTM-2012-32542es
dc.relation.publisherversionhttp://dx.doi.org/10.1016/j.laa.2015.06.031
dc.rights© 2015 Elsevieren
dc.rightsAtribución-NoComercial-SinDerivadas 3.0 España*
dc.rights.accessRightsopen access
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/es/*
dc.subject.ecienciaMatemáticases
dc.subject.otherSymmetric matrix factorizationen
dc.subject.othersymmetrizeren
dc.subject.othersymmetrizer computationen
dc.subject.othereigenvalue methoden
dc.subject.otherlinear equationen
dc.subject.otherprincipal subspace computationen
dc.subject.othermatrix optimizationen
dc.subject.othernumerical algorithmen
dc.subject.otherMATLAB codeen
dc.titleComputing matrix symmetrizers. Part 2: new methods using eigendata and linear means; a comparisonen
dc.typeresearch article*
dc.type.hasVersionAM*
dspace.entity.typePublication
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