Representations Of The Drazin Inverses For Some Block Matrices

When we solve singular differential and difference equations,the Drazin inverse of block matrices play an important role in it,also widely used in other areas,such as in Markov chains,iterative methods.For example,for the second order differential equation Ex(t)+Fx(t)+Gx(t)=0when E is singular, the solution is where z(t)=e-λtx(t),E2=(λ2E+λF+G)-1(2λE+F),F2=(λ2E+λF+G)-1E. Let we name S=D-CADB the generalized Schur complement of M,where AD is the Drazin inverse of A.The Schur complement S=D-CA-1B plays an important role in matrix analysis and linear matrix inequalities.So in chapter3,we first give the formula of the Drazin inverse of additive matrices P+Q(P,Q∈□n×n)under one of the following conditions:(3.1)QPQ+QP2+P2Q+P3=0,QPQ2=0;(3.2)P2Q=0,Q2P=0,PQPQ=0, then using the above results,we give the formula of the Drazin inverse of M which the generalized Schur complement is nonsingular under one of the following conditions:(3.3)BCAπA=0,BCAπB=0,DCAπA=0,DCAπB=0:(3.4)ABCAπ+BDCAπ=0,ABCAπ+D2CAπ=0,CAπBCAπ=0, where the result(3.3)in the above new results develop the previous results.In Chapter4,we give the formula of the Drazin inverse of anti-triangular matrices under one of the following conditions respectively:(4.1)BCAAD=0,AπBC=0;(4.2)BCADB=0,BCAπ=0(CADBC=0,AπBC=0);(4.3)CABC=0,ABCAπ=0,A2BC=0;(4.4)ADBC=0,ABCAπ=0,where the result(4.3)and(4.4)in the above new results develop the previous conclusions.