The Nature Of The Non-negative Matrix And Random Matrix Norm

Firstly, the properties of nonnegative matrices, irreducible matrices are explored. Secondly, norms of stochastic matrices are discussed. Thirdly, two important problems about consimilar and the solution of the equation AX-XB=C are further investigated.This paper contains four chapters, we give it as follows:In chapter one, some notations, fundermental definitions are introduced and some theorems are given. In part one, we introduce the definitions of the nonnegative matrix, irreducible matrix, stochastic matrix and so on.In part two, we provide some theorems about them.In chapter two, starting from foundations, several known propositions and theorems are further explored, then some properties of the irreducible matrix are concluded by means of constructing a new matrix ect. Last, the properties of idempotent matrix in special form are researched.In chapter three, bounds of (?)·(?)1,(?)·(?)2,(?)·(?)p to stochastic matrices are discussed; also the sufficient and necessary conditions are given when those bounds are attained.In chapter four, in the first part, the properties of AA are studied. In the second part, linear transformations are introduced to demonstrate the solution of the equation AX-XB=C and an application is given out.