Some Discussions On The Drazin Inverses Of Combinations Of Two Idempotent Matrices

In this paper,by using the properties of matrix zero space,and the properties of idempotent matrix,the group inverse,the definition and research method of undetermined coefficients of Drazin inverse some combination of two different idempotent matrix under the condition of different group of inverse,Drazin inverse formula and index.These results generalize the combination of two different idempotent matrix in the special conditions of the corresponding results.This paper mainly studies the following contents:(1)By using the properties of null space of matrices,the rank of the combinationsα1P+b1Q+a2PQ+b2QP+…+a2n-1(PQ)n-1 P+b2n-1(QP)n-1Q+a2n(PQ)n of two different nonzero idempotent matrices P and Q over the complex filed C,where a1,b1,…a2n∈C,a1,b1≠0,was proved to be independent with the choice of its cofficients and under the condition(QP)n=0.Therefore,the existence of the group inverse of the combination was also obtained.At the same time,the formula for the group inverse of the combination aP+bQ+cPQ+dQP was presented under the condition(QP)n=0 and(PQ)n=(QP)n.(2)By using the properties of idempotent matrix,the definition of group inverse and Drazin inverse and the undetermined coefficient method,the formula for group inverse,Drazin inverse and the index of some combination are obtained under the condition(PQ)2P=(PQ)2,(PQ)nP=(PQ)n,(PQ)n+1=(PQ)n.(3)We study the dimension of algebras alg(P,Q)under the condition(PQ)m+1=(PQ)m(m≥1).At this moment,alg(P,Q)is spanned by P,Q,PQ,QP,…,(PQ)m P,(QP)mQ,(QP)m+1.