The Drazin Inverse Of Combination Of Two Idempotents

Duing to the Drazin inverse of matrices plays an important role in research branch of generalized inverse matrix, Drazin inverse has been widely used in many areas. It has attracted many domestic and foreign scholars who have obtained many results for operator combination’s Drazin inverse expression under different conditions in recent years.In the first chapter, we simply introduce generalized inverse matrix, especially the research background, meaning and research status at home and abroad of the Drazin inverse.In the second chapter, we firstly introduce the basic concepts of Drazin inverse, then discuss (aP-bQ)D under the case(PQ)2=PQ. We describe (aP-bQ)D by using the combination of matrices P,Q, PQ, QP, PQP, QPQ and(QP)2, and the results of the index about the matrices are acquired.In the third chapter, we gain the Drazin inverse and the index of nonlinear combination (aP+bQ-cPQ) and (aP+bQ-cPQ-dQP). Thus the conclusions of this paper generalize the relative results of Deng, Zhang and Wu, Xie and Zuo.In the fourth chapter, firstly, a summary of the work for this article is given. Secondly, we will think about the next step of research contents. Because the condition (PQ)2=PQ can reduce the condition (PQ)3=PQ , but the converse is wrong. We can discuss and index under(PQ)3=PQ. A more general case (PQ)k=PQ(k>3) can also be considered, so are their index.