The Drazin Inverse For The Sum Of Two Matrices And Its Applications Over Skew Fields

Because of the Drazin inverse of the matrix plays an important role in solving singular differential equation and difference equation, iterative method, Markov chain, cryptography and many other important areas, the Drazin inverse of the matrix is widely accepted as a very important branch of Generalized inverse matrix and attracted more attention of scholars. Thus promote the development of the research about the theory and application of the Drazin inverse. More and more results on the Drazin inverse of block matrix were given. In1977, CD. Meyer used limit method to give the Drazin inverse of triangular matrix over the complex fields, till now, many scholars give the explicit representations for the Drazin inverse of block matrix under some conditions. Only a few of the results are given over skew fields. In this paper, all of the new results are given in skew fields, and promote some results over complex fields.In chapter1, we briefly introduce on the research background, meaning and the state of research of the generalized inverse of matrix. The chapter2introduces the related concepts of the generalized inverse and the application of theory of generalized inverse used in solving some specific linear equations. In chapter3, we give the explicit representations of the Drazin inverse for the upper triangular matrix over skew fields, then give the additive results of the Drazin inverse over skew fields when one of the following conditions is satisfied:(1) PQ=0;(2) P2QP=0,P3Q=0,Q2=0;(3)PQP2=0, QP3=0, Q2=0As the application of formulas, in chapter4, we give the following conclutions:(i) For block matrix M=(?)(where A is square and D-CADB=0) over skew fields, we give the explicit representations of MD when one of the following conditions is satisfied:(1) A2BC=0,ABCA=0and ABCB=0;(2) BCA2=0, ABCA=0and CBCA=0;(3) BCA" is rnilpotentand (I+BC(AD2)ABCAπ=0;(4) CAπBC=0and AA#BC=0;(5) CAπBCAπ=0and (A+BCAD)BCAπ=0;(6) AπBC is p nilpotentand (A+ADBC)A"BC=0. (ⅱ)For block matrix (?)(where A and0are square)over skew fields give the explicit representations of M D when one of the following conditions is satisfied:(1)A2BC=0,ABCA=0and ABCB=0:(2)BCA2=0,ABCA=0and CBCA=0:(3)BCAπ=0and BCADB=0:(4)AπBC=0and CADBC=0:(5)ABCAπ=0,A.BC=0and CABC=0.In this paper,we also give some examples to vetify the coffeetness of the conelutions for the new results given above.