The Operator Method For The Generation And Factorization Of Circulant Matrices

In this thesis, we use the shift operator method to study the structure andproperties of circulant matrices. From the view of linear operator, the intrinsicconnections between circulant matrices and other related structured matrices arederived, meanwhile three different kinds of generation ways of circulant matricescorresponding to the shift operator expression are discussed. With the relationshipbetween circulant matrices and structured matrices, such as Vandermonde matrices,Toeplitz matrices and Toeplitz-Bezoutions, one can study the operator method forfactorization and reduction of circulant matrices. After that we summarize thegeneration ways of generalized circulant matrices, such as the block circulantmatrices, r-circulant matrices and anti-circulant matrices. The properties of blockcirculant matrices are proved by the operator method, and then the media rolebetween Toeplitz matrices and Toeplitz-Bezoutions that r-circulant matrices arediscussed. All of them have some applications in the control system theory.The present thesis is divided into five parts:In chapter one, the history and present research of structured matrices andlinear operators are introduced, and the main work is presented.In chapter two, in the first and second parts we introduce the conception andproperties of Toeplitz-Bezoutions and related matrices, the definition of linearoperators, and the linear operator representation of some kinds of structuredmatrices. The third part is dedicated to discuss the operator method for studying thegeneration ways of circulant matrices, respectively, and to illustrate the relationshipbetween circulant matrices and the basic circulant matrices, Toeplitz matrices andToeplitz-Bezoutions. We introduce three different kinds of generation ways ofcirculant matrices corresponding to the shift operator expression.In chapter three, two kinds of factorization ways of circulant matrices areconsidered：the Vandermonde factorization and the Barnett factorization. By therelationship between circulant matrices and Vandermond matrices，one can get theVandermonde reduction of circulant matrices，and deduce the relationship with theVandermonde factorization.In chapter four，the circulant matrices are extended to the study of generalizedcirculant matrices. We first reprove the properties of block circulant matrices by operator method, which illustrate the block circulant matrices are the generalizationof the common circulant matrices, so we get some similar properties and linearoperator expressions. Finally the generation way of r-circulant matrices includinganti-circulant matrices is discussed, which extands the connection between Toeplitzmatrices and Toeplitz-Bezoutions.The last chapter is used to summarize the main work of the paper and put someunsolved problems for further research.