Research On Some Problems Of Bezout Matrix

The concept of Bezoutian and its various generalizations play an impor-tant role in modern linear algebra. And this phenomenon has attracted manyscholars' attention and lots of available and important results have been ob-tained.More recently Bezoutians appeared in connection with realization the-ory, the Cauchy index and stability theory. In particular, operator theorythe inversion of Bezoutian and its various generalizations (Hankel,Toeplitzmatrics)play an essential role. So, it is important practical meanings to studyBezoutians and its inversion.In this paper,the centrosymmetric Bezoutians,Bezoutians space and itsinversion are discussed.This thesis is composed of four chapters, which mainlystudy the dissolution of H-Bezoutians generated by symmetric polynomials(T+H-Bezoutians),the construction of Bezoutians space and the conditions suchthat the inversion of Bezoutians being theirselves.in chapter one,the backgrounds and significance of the problems are given.The main work done in this paper is introduced.Chapter two mainly considers the dissolution of H-Bezoutians generatedby symmetric polynomials as well as T+H-Bezoutians.And the relation be-tween them are discussed.In chapter three, we consider two types of Bezoutians and give the equiva-lent conditions of generating the linear Bezoutians space.It is proved that anysymmetric matrix can be linearly represented by some Bezoutians.In the last chapter, by using operator theory, we obtain the necessary andsufficient condition for which the inversion of Bezoutians are theirseIves. Thespecific construction of this kind of matrices is derived.