toeplitz and circulant matrices a review

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1、ToeplitzandCirculantMatrices:AreviewToeplitzandCirculantMatrices:AreviewRobertM.GrayDeptartmentofElectricalEngineeringStanfordUniversityStanford94305,USArmgray@stanford.eduContentsChapter1Introduction11.1ToeplitzandCirculantMatrices11.2Examples51.3GoalsandPrerequisites9Chapter2TheAs

2、ymptoticBehaviorofMatrices112.1Eigenvalues112.2MatrixNorms142.3AsymptoticallyEquivalentSequencesofMatrices172.4AsymptoticallyAbsolutelyEqualDistributions24Chapter3CirculantMatrices313.1EigenvaluesandEigenvectors323.2MatrixOperationsonCirculantMatrices34Chapter4ToeplitzMatrices37vviC

3、ONTENTS4.1SequencesofToeplitzMatrices374.2BoundsonEigenvaluesofToeplitzMatrices414.3BandedToeplitzMatrices434.4WienerClassToeplitzMatrices48Chapter5MatrixOperationsonToeplitzMatrices615.1InversesofToeplitzMatrices625.2ProductsofToeplitzMatrices675.3ToeplitzDeterminants70Chapter6Appl

4、icationstoStochasticTimeSeries736.1MovingAverageProcesses746.2AutoregressiveProcesses776.3Factorization80Acknowledgements83References85Abstractt0t−1t−2···t−(n−1)t1t0t−1.t2t1t0........tn−1···t0Thefundamentaltheoremsontheasymptoticbehaviorofeigenval-ues,inverses,andpro

5、ductsofbandedToeplitzmatricesandToeplitzmatriceswithabsolutelysummableelementsarederivedinatutorialmanner.Mathematicaleleganceandgeneralityaresacrificedforcon-ceptualsimplicityandinsightinthehopeofmakingtheseresultsavail-abletoengineerslackingeitherthebackgroundorendurancetoattackthe

6、mathematicalliteratureonthesubject.Bylimitingthegeneralityofthematricesconsidered,theessentialideasandresultscanbecon-veyedinamoreintuitivemannerwithoutthemathematicalmachineryrequiredforthemostgeneralcases.Asanapplicationtheresultsareappliedtothestudyofthecovariancematricesandtheir

7、factorsoflinearmodelsofdiscretetimerandomprocesses.vii1Introduction1.1ToeplitzandCirculantMatricesAToeplitzmatrixisann×nmatrixTn=[tk,j;k,j=0,1,...,n−1]wheretk,j=tk−j,i.e.,amatrixoftheformt0t−1t−2···t−(n−1)t1t0t−1.Tn=t2t1t0...(1.1)......tn−1···t0Suchmatricesariseinman

8、yapplications.Forex