Galba, E. F.; Deineka, V. S.; Sergienko, I. V. Weighted pseudoinverses and weighted normal pseudosolutions with singular weights. (Russian, English) Zbl 1199.65130 Zh. Vychisl. Mat. Mat. Fiz. 49, No. 8, 1347-1363 (2009); translation in Comput. Math., Math. Phys. 49, No. 8, 1281-1297 (2009). Summary: Weighted pseudoinverses with singular weights can be defined by a system of matrix equations. For one of such definitions, necessary and sufficient conditions are given for the corresponding system to have a unique solution. Representations of the pseudoinverses in terms of the characteristic polynomials of symmetrizable and symmetric matrices, as well as their expansions in matrix power series or power products, are obtained. A relationship is found between the weighted pseudoinverses and the weighted normal pseudosolutions, and iterative methods for calculating both pseudoinverses and pseudosolutions are constructed. The properties of the weighted pseudoinverses with singular weights are shown to extend the corresponding properties of weighted pseudoinverses with positive definite weights. Cited in 14 Documents MSC: 65F20 Numerical solutions to overdetermined systems, pseudoinverses 15A09 Theory of matrix inversion and generalized inverses Keywords:weighted pseudoinverses with singular weights; weighted normal pseudosolutions; matrix power series; matrix power products; iterative methods PDFBibTeX XMLCite \textit{E. F. Galba} et al., Zh. Vychisl. Mat. Mat. Fiz. 49, No. 8, 1347--1363 (2009; Zbl 1199.65130); translation in Comput. Math., Math. Phys. 49, No. 8, 1281--1297 (2009) Full Text: DOI