The Expression Of The Drazin Inverse Of Sums And Differences Of Two Matrices And Its Application

In this note, we mainly investigate the representations of the Drazin inverse of sums and differences of two matrices and its applications. The paper is organized as follows:In chapter 1, we give some required preparative knowledge including general sym-bols, definitions and basic lemmas. Also, the main results of the note are presented.In chapter 2, we present the relations between (PQ)d,(PQP)d, PQd, QdP and P, Q, Pd, Qd, and explicit representations of (P±Q)d, (P±PQ)d of two matrices P and Q, as a function of P, Q, Pd, Qd, under the conditions P3Q= QP and Q3P= PQ.In chapter 3, we give explicit expressions of (P±Q)d of two matrices P and Q, in terms of P, Q, Pd and Qd, under the condition that PQ= P2, and apply the result to finding an explicit representations for the Drazin inverse of a 2×2 block matrix (?) under some conditions.