Matrix Of Exchangeability

Special matrices play an important role in matrix analysis and matrix computation and have wide applications in computational mathematics, economics, physics, biology, applied mathematics and etc. Great progress obtained in the researches on special matrices will give improvements in computational mathematics. With the applications of matrices are more and more abroad, the commutativity of matrix is more and more recognition by scholar and technology worker. The commutativity of matrix not only plays an important part in the matrix computation, but also in the secondary planet, communcation and other fields.In this thesis, we focus on the most matrices that can not be satisfying the commutativity. We give some quality and sufficiency conditions for some special matrices, scalar matrices and some upper triangle matrices. Also by the use of the Frobenius standard pattern and the combination of sign pattern matrix, we get more precise conditions of the commutant of the full pattern.There are three chapters in this paper:In chapter 1, we introduce the background of the commutant about the matrix with some concepts and notation related.In chapter 2, by the study of some special formula, matrix such as, diagonal matrix upper triangle matrix etc., we get some sufficiency conditions for the commutant of matrix, and some beautiful quality about commutativity.The chapter 3 is the main results of this paper which are based on Johnson and Graca. Marquis's swath conditions theorem. By the use of the Frobenius standard pattern and the combination qualities of sign pattern matrix, we get more precise conditions about the commutant of the full pattern. That is by we discuss the positions of the nonzero entries in the subpatterns further and obtain that if there are two nonzero entries among the subpatterns, then the two entries must be in a block, and if there are three or four nonzero entries among the subpatterns what conditions they should satisfy. Furthermore we get the conclusion that if there are more nonzero entries among the subpatterns what conditions they should satisfy, or not it must not be commutting with the full pattern.