Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

Abstract

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive semidefinite weights and on the construction of iterative methods and regularized problems for the calculation of weighted pseudoinverses and weighted normal pseudosolutions based on these representations and expansions. The use of these methods to solve constrained least squares problems is examined.

Existence and uniqueness theorems in the theory of weighted pseudoinverses with singular weights

“…THEOREM 6. For system (3) to have a unique solution X A BC = + , it is necessary and sufficient that…”

“…As follows from (1), the matrix AX is left-symmetrizable by the symmetrizer B, and the matrix XA is right-symmetrizable by the symmetrizer C. Some studies (for example, [2,3] and the bibliography therein) consider weighted pseudoinverses with singular weights defined by system (1) under some existence conditions for its solutions. These papers analyze the properties of such matrices and weighted normal pseudosolutions, obtain and study the representations and expansions of matrices, and construct iterative methods and regularized problems to find approximations to weighted pseudoinverses and weighted normal pseudosolutions with singular weights.…”

“…As above, we will assume that B and C are symmetric positive semidefinite matrices. We will pay much attention to system (3) to determine the necessary and sufficient conditions whereby a unique solution of this system of matrix equations exists and will analyze the existence of a unique weighted normal pseudosolution determined based on a weighted pseudoinverse with singular weights. If B C E = = , where E is a unit matrix, the system of matrix equations (3) defines the Moore-Penrose pseudoinverse [5,6] of the matrix A; we denote it by A EE + .…”

“…In Sec. 2, we will establish the conditions for system (3) with singular weights B and C to have a unique solution and will represent weighted pseudoinverses with singular weights in terms of coefficients of characteristic polynomials of symmetrizable and symmetric matrices. In Sec.…”

Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights

“…It follows from (1) that the matrix AX is symmetrizable from the left by the symmetrizer B, and the matrix XA is symmetrizable from the right with the symmetrizer C. There are many works (see [2] and the bibliography therein) devoted to weighted pseudoinverse matrices with singular weights defined by conditions (1) and (2), in which properties of weighted pseudoinverse matrices and weighted normal pseudosolutions are investigated, representations and expansions of weighted pseudoinverse matrices are obtained and investigated, and iterative methods and regularized problems for the calculation of approximations of these weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights are constructed. In particular, a representation of the weighed pseudoinverse matrix defined by conditions (1) and (2) in terms of the coefficients of the characteristic polynomial of a symmetrizable matrix and its limit representation were obtained in [3]. Various representations of the matrix A BC + defined by conditions (1) and (2) were obtained in [4], and its expansions in series and limit polynomial representations were obtained in [5] and [6].…”