Galba E. F.

We present the definition of weighted pseudoinverse matrices with nonsingular indefinite weights and study these matrices. The theorems on existence and uniqueness for these matrices are proved. Weighted pseudoinverse matrices with indefinite weights are represented in terms of the coefficients of characteristic polynomials of symmetrizable matrices. The decompositions of weighted pseudoinverse matrices into matrix power series and products and their limit representations are obtained. We also propose regularized iterative methods for the determination of these matrices.

A weighted singular-valued decomposition of matrices with singular weights is obtained by using orthogonal matrices. The necessary and sufficient conditions for the existence of the constructed weighted singular-valued decomposition are established. The indicated singular-valued decomposition of matrices is used to obtain a decomposition of their weighted pseudoinverse matrices and decompose them into matrix power series and products. The applications of these decompositions are discussed.

For one of definitions of weighted pseudoinversion with singular weights, necessary and sufficient conditions for the existence and uniqueness are obtained. Expansions of weighted pseudoinverse matrices in matrix power series and matrix power products are obtained. Relationship is established between the weighted pseudoinverse matrices and the weighted normal pseudosolutions. Iterative methods for the calculation of both weighted pseudoinverse matrices and weighted normal pseudosolutions are constructed.

We obtain expansions of weighted pseudoinverse matrices with singular weights into matrix power products with negative exponents and arbitrary positive parameters. We show that the rate of convergence of these expansions depends on a parameter. On the basis of the proposed expansions, we construct and investigate iteration methods with quadratic rate of convergence for the calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions. Iteration methods for the calculation of weighted normal pseudosolutions are adapted to the solution of least-squares problems with constraints.

On the basis of the Euler identity, we obtain expansions for weighted pseudoinverse matrices with positive-definite weights in infinite matrix power products of two types: with positive and negative exponents. We obtain estimates for the closeness of weighted pseudoinverse matrices and matrices obtained on the basis of a fixed number of factors of matrix power products and terms of matrix power series. We compare the rates of convergence of expansions of weighted pseudoinverse matrices in matrix power series and matrix power products to weighted pseudoinverse matrices. We consider problems of construction and comparison of iterative processes of computation of weighted pseudoinverse matrices on the basis of the obtained expansions of these matrices.

For a rectangular real matrix, we obtain a decomposition in weighted singular numbers. On this basis, we obtain a representation of a weighted pseudoinverse matrix in terms of weighted orthogonal matrices and weighted singular numbers.

Weighted pseudoinverse matrices with singular weights are represented in terms of the coefficients of characteristic polynomials of certain square matrices. By using the expression obtained, we construct limit representations of weighted pseudoinverse matrices with singular weights.