Additional results on index splittings for Drazin inverse solutions of singular linear systems (Q2717794)

The so-called proper splitting with \(R(U)=R(A)\) and \(N(U)=N(A)\), where \(A\) is an \(n \times n\) singular matrix \(A \in \mathbb{R}^{n \times n}\), \(R(A)\) denotes the range of \(A\), \(N(A)\) denotes the null space of \(A\), which was introduced by \textit{A. Berman} and \textit{R. J. Plemmons} [SIAM J. Numer. Anal. 11, 145-154 (1974; Zbl 0273.65029)], is generalized to the index splitting of \(A\) with \(Ind(A)=k\) of the form \(A=U - V\), where \(R(U)=R(A^k)\) and \(N(U)=N(A^k)\). The index splitting is analyzed with the aim to replace the Moore-Penrose inverse \(A^+\) and \(A^{-1}\) in the iterative scheme \(u^{(i+1)}=U^+Vu^{(i)} + U^+f\) when solving a singular system of linear equations \(Au=f\), where \(A \in \mathbb{R}^{n \times n} \) and \(u, f\) are vectors in \( \mathbb{R}^n\). NEWLINENEWLINENEWLINEThe new replacing inverse, denoted \(A^D=X\), satisfying equations \(AX=XA\), \(A^{k+1}X=A^k\) and \(AX^2=X\), is called the Drazin inverse of \(A\). When \(Ind(A)=1\), the matrix \(X\) is called the group inverse of \(A\) and denoted by \(X=A^{\#}\). The previously achieved results showing that the iterations \(u^{(i+1)}= U^{\#}Vu^{(i)}+U^{\#}f\) converge to the Drazin inverse solution \(A^Df\) of the linear system if and only if the spectral radius \(\rho(U^{\#}V)\) is less than one are further extended. NEWLINENEWLINENEWLINEThe characteristics of the Drazin inverse solution \(A^Df\) are given and equivalent conditions for the convergence of iterations are presented. Conditions are also given under which the iterations based on the index splitting are of the monotone type. Some comparison theorems on the index splitting of the same matrix \(A=U_1-V_1 = U_2-V_2\) are also discussed. Finally a characterization of the iteration matrix \(T=U^{\#}V\) of the index splitting is presented.