Nonnegative Matrix And Operator Equation

In this paper, nonnegative matrix and operator equation have been discussed. The study of nonnegative matrix theory began earlier in 20th century. With the fast development of the theory, now it has become a hot branch playing the role of an initiator in morden mathematics. It has unexpected relations and interinfiltrations with physics, linear system, econimic, probability and statistics, indeed social science as well as some other important branches of mathematics. In recent years, many scholars both here and abroad have focused on the theory of nonnegative matrix and have introduced more and more new methods. For example, completely positive matix, totaly negative matrix, copositive matrix and so on were introduced successively. In present, these matrices have become important tools in studying nonnegative matrix.Operator equitation has always been one of the more active branch in mathematical research. For operator equation, it is interesting for people to solve the equation and study the properties of the solution. These have the vital signifinance in the solution actual problem.The paper is divided into four chapters.In chapter 1, some notations and definitions are introduced and some well-known theorems are given. In section I, we give some technologies and notations, and introduce the definitions of irreducible nonnegative matrix, permutation matrix, peripheral spectrum, spectrical subset and bounded linear operators and so on. In section II, we give some well-known theorems, such as distinguished Perron-Frobenious Theory.In chapter 2, we first study the matrix which possessses the Perron-Frobenious property. For A ∈ R^n×n, we establish three equivalent properities as follows:(i) Both matrices A and A^T posses the strong Perron-Frobenius property;(ii) A is an eventually positive matrix;(iii) A^T is an eventually positive matrix.Secondly, we discuss some properties of matrix above which are similiar to the irreducible nonnegative matrix. Lastly, we study the pertubation of matrix which possesses the Perron-Frobenious property.In chapter 3, we first discuss the completely positive matrix and give a sufficient and neessary condition of whence a double nonnegative matrix is a completely positive matrix, i.e., For A = (o#) 6 DNNn and suppose that A = T^T where T is a real kxn matrix and k = rank(A). Then A G CPn if and only if there exist m nonzero vectors X\, ? ? ? ,xm € S? such that\-------h xmxTm - IkFurthermore, the smallest such number m is equal to cprank(A). In the sequel, we get an equivalent condition of A = An, where A is an reducible nonnegative matrix, i.e., If A > 0 is reducible and r(A) = 1, then A — An = 0 if and only if there exists a permutation matrix P such thatPAPT =An 000 00 0Arrwhere An is irreducible and satisfies A^ = Aa{\ < i < r).In chapter 4, we study the operator equation XA — AX = Xp, for all positive integers 1 < p < oo. We get two main results: (1) The solutions of XA — AX = Xp are quasinilpotent. As a corollary, the solution X is nilpotent if the dimension of H is finite (in [32]);(2) If A" is a nilpotent solution of operator equation XA—AX = Xp for all positive integers p > 2, then XEa(