Research On The Properties Of Several Types Of Special Matrices

The theories about special matrices have a fairly wide application in the field of engineering computing,automatic control,system identification,numeri-cal analysis,optimization theory and so on.The study about the properties of special matrices has become a hot issue in matrix theory and numerical algebra in the modern mathematics,and got more and more attention among scholars.Therefore,the study on special matrices will be expected to received many mean-ingful conclusions.On the basis of a large number of literature,this paper studied some properties and characteristics of some special matrices deeply,which can be divided into the following five chapters:In the first chapter three kinds of special matrices were mainly be introduced,they are anti-centrosymmetric matrix,idempotent matrix and skew-symmetric and skew-circulant matrix.The research background,research significance,re-search status and the basis framework of this paper were also given in this chapter.In the second chapter,the properties of anti-centrosymmetric matrices were researched.According to the structural characteristics of the anti-centrosymmetric matrix,some new methods were used to prove the necessary and sufficient condi-tions of a matrix being anti-centrosymmetric and its properties of eigenvalue and eigenvector.In addition,the nonsingularity of the anti-centrosymmetric matrices were discussed,that the odd order anti-centrosymmetric matrix is singular was obtained,and two methods of computing inverse of the matrices?even order?were given.In the third chapter,idempotent matrices over skew field were researched and some properties of idempotent matrices were extended from general complex domain to skew field.In this chapter,the following conclusions were obtained:?1?the four equivalent conditions of idempotent matrices over skew field;?2?the necessary and sufficient conditions which could infer that the linear combinations A₁ + A₂ and A₁-A₂ of idempotent matrices over skew field Ai,A₂ were also idempotent matrices;?3?the necessary and sufficient conditions of nonsingularity of correlative left linear combinations c₁A₁-c₂A₂ and c₁A₁A₂ + c₂A₂A₁?where C1,c₂ ? KA₁,A₂?of idempotent matrices over skew field A₁,A₂.In the fourth chapter,some properties of skew-symmetric and skew-circulant matrix were extended from general complex domain to skew field.The relation-ships between skew-symmetric and skew-circulant matrix and symmetric circulant matrix,symmetric and skew-circulant matrix and skew-circulant matrix,over skew field,were obtained.Meanwhile,the linear expression of skew-symmetric and skew-circulant matrix under fundamental skew-circulant matrix was also obtained.In addition,over skew field,the sufficient condition,which could infer that one ma-trix was skew-symmetric and skew-circulant matrix and that skew-symmetric and skew-circulant matrices were commutative,as well as some properties of inverse matrix of skew-symmetric and skew-circulant matrix were acquired.In the fifth chapter,the content of this paper was summarized,and the future research direction was prospected.