Representations for the Drazin inverse of the sum \(P+Q+R+S\) and its applications (Q958035)

Let \(P\), \(Q\), \(R\) and \(S\) be complex square matrices and \(M=P+Q+R+S\). A quadruple \((P,Q,R,S)\) is called a pseudo-block decomposition of \(M\) if \[ PQ=QP=0,\quad PS=SQ=QR=RP=0, \quad\text{ and }\;R^D=S^D=0 \] where \(R^D\) and \(S^D\) are the Drazin inverses of \(R\) and \(S\), respectively. The authors investigate the problem of finding explicit representations for the Drazin inverse of \(M\) and \(P+Q+R\).