Weighted pseudoinverses and weighted normal pseudosolutions with singular weights

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.