Universal iterative methods for computing generalized inverses
Let \(H_1,H_2\) be two finite-dimensional complex Hilbert spaces and \(A:H_1\to H_2\) a linear operator. Consider the following conditions on an operator \(X\) from \(H_2\) into \(H_1\): (1) \(AXA=A\); (2) \(XAX=X\); (3) \((AX)^\ast=AX\); (4) \((XA)^\ast=XA\); (5) \(AX=XA\). For \(\mathcal S\) a subset of \(\{1,2,3,4,5\}\), let us call an operator \(X\) satisfying the conditions in \(\mathcal S\) an \(\mathcal S\)-inverse of \(A\). If \(\mathcal S=\{1,2,3,4\}\) or \(\mathcal S=\{1,2,5\}\), the corresponding \(\mathcal S\)-inverses are unique and coincide with the Moore-Penrose inverse and the group inverse of \(A\), respectively. The authors construct iterative processes, based on hyper-power iterative method or on Neumann-type series expansions, which converge to a \(\{1,2\}\), a \(\{1,2,3\}\), a \(\{1,2,4\}\), the Moore-Penrose, the group, and the weighted Moore-Penrose inverse, respectively (the last is defined by replacing (3) and (4) with \((MAX)^\ast=MAX\) and \((XAN)^\ast=XAN\), where \(M,N\) are given positive definite ``weight'' matrices). In all cases, it is assumed that certain operators \(W_1,W_2\) are known such that \(W_2 A W_1\) is invertible. Error bounds are also established and it is shown that the methods are self-correcting. The paper concludes with a few examples illustrating the theory.
finite-dimensional complex Hilbert spaces
scientific article; zbMATH DE number 1225610