Abstract. Two variants of weighted singular value decompositions of matrices with singular weights using weighted orthogonal matrices are obtained and analyzed. Based on this singular value decompositions of matrices, decomposition of weighted pseudoinverse matrices with singular weights and decomposition of these matrices into matrix power series and products are obtained. The application of these decompositions is determined.

Keywords: weighted singular value decomposition of matrices with singular weights, weighted pseudoinverse matrices with singular weights, weighted normal pseudosolutions, weighted orthogonal transformations, matrix power series, matrix power products, regularized problems.