Some Classes Of Partial Ordering Based On Generalized Inverses

As generalized inverse theory plays an indispensable role in many fields such as differential equation,numerical calculation and optimization.As a result,more and more experts and scholars have been attracted to study it deeply from the perspective of complex matrix,bounded linear operators in Hilbert space and rings and many research results have been obtained.This paper mainly discusses the necessity and sufficiency of the existence of MP inverse,group inverse,core inverse,pseudo-core inverse,(B,C)-inverse of a class of block matrices over rings and the partial ordering based on these generalized inverses.The necessary and sufficient conditions for whether two elements constitute partial ordering are given.Finally,this paper studies the partial order maximal element under the ? order(where ? is some generalized inverse)and maximal element of partial order based on MP inverse,group inverse and core inverse of a class of block matrices over rings.It mainly consists of two parts.In the first part,we mainly study the necessary and sufficient conditions for the existence of MP inverse,group inverse,core inverse,pseudo-core inverse,(B,C)-inverse of a class of block matrices A= U(?)U*over rings(where U is a unitary matrix)and partial ordering based on MP inverse,group inverse and core inverse.By Hartwig-Spindelbock decomposition of com-plex matrices and matrices over quaternion division ring,this kind of matrix generalizes all n-order complex matrices and matrices over quaternion division rings.Firstly,we prove that At exists if and only if {A1A1*+A2A2*)t and A(1,4)exist by the inclusion relation of the range,in this case,A(?)=U(?)U*.Secondly,we discuss the(B,C)-inverse of A.When A=U(?)U*,B=U(?)U*,C=U(?)U*(orC=U(?)U*),We give the necessary and sufficient conditions and expressions for the existence of(B,C)-inverse of A.In partic-ular,we obtain the corresponding result of the group inverse(core inverse)of A.Finally,we study the successors of A under*,#,(?)partial orders and prove that if A1 is reversible,then A?*B if and only if B=U(?)U*,where Y,Z meet YA1*+ZA2*=0.A?#B if and only if B=U(?)U*,where B12,B22 meet A1B12+A2B22=AlA2.A?(?)B if and only if B=U(?)U*.In the second part,we study the maximal element of the ? order(where ? denotes some generalized inverse).when m?=m+,that is minus partial order,when m?=m(?),that is(?)partial order,when m?=m#,that is#partial order,when m?=m(?),that is(?)partial order,? unifies the classical generalized inverses.At the same time,we study the necessary and sufficient conditions for A to be the maximal element under the partial order of#,(?),(?).When m??m{2},we give the expression of successors under the ? order and the necessary and sufficient conditions of maximal elements,which generalizes Guo Jinbao[47]related conclusions on minus partial ordering.When ??{#,(?)},we prove that m is a maximal element under the ? order if and only if m is unilateral reversible if and only if m is reversible.When R is a semiprime ring.We give the relationship between maximal elements and annihi-lators and prove that if ??{#,(?)},then m is a maximal element under the ? order if and only if the annihilator of m is zero.At the same time,it is proved that if m??{1,2} and m is a maximal element under the ? order,then m is invertible if and only if m is group invertible.when R is a*unitary regular ring,We give the relation between maximal element and reversible element and discuss the problem that the product of maximal element is maximal element.Finally,we give the necessary and sufficient condition for A to be the maximal elements under*((?))partial order in the idempotent matrix set.