Some Studies On Minus Partial Order Of Matrices Over Bezout Domain

since the 1980s, the theory and application of minus partial order on matrices has been paid much attention. This paper mainly discusses the theory of minus partial order on matrix over ring. Let R be a ring and 0≠A € Rm×n. Then there exists a smallest positive integer r such that A=BC, where B ∈Rm×n, C ∈ RT×n. the r is called the inner rank of A and denoted by p(A). In particular, when A=0, ρ(A)=0. Let A,B∈Rm×n,if ρ(B-A)=ρ(B)-ρ(A), then A is said to be below B, and denoted by A≤B.The binary relation"≤" is called minus partial order, it’s a kind of partial order.This paper is divided into three chapters. The first chapter introduces the subject background, the main research contents and conclusions.The second chapter studies the minus partial order on matrices over rings and Bezout domain. First, we point out the fundamental properties of minus partial order on matrices over rings. For example:A≤B(?)PAQ≤PBQ, where P,Q are invertible matrices. Then, we prove some important properties over Bezout domain. For example, if A≤B, then B is idempotent(?)A is idempotent; if B has a generalized inverse, then A has a generalized inverse, too. This chapter mainly proves a series of equivalent conditions of the minus partial order over Beozut domain, including:· A≤B and B has a generalized inverse.·If B-is a generalized inverse of B, then A=AB-B=BB-A=AB-A.· A has a generalized inverse, and{B-}(?){A-}, i.e., each generalized inverse of B is also a generalized inverse of A.·There exists an M ∈ Rn×m such that B-A=BMB and MBM=M.· There exists a minimum rank decomposition of B=PiQi, such that A=P1TQ1 where T is a idempotent matrix.In the last section of this chapter, we prove five basic equivalent conditions of minus partial order on matrices over division rings.The third chapter, we make a preliminary discussion of algebra characterization about minus partial order automorphism on matrices over division rings.