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

For one definition of weighted pseudoinversion with singular weights, we establish necessary and sufficient conditions for the existence and uniqueness of a solution of a system of matrix equations. Expansions of weighted pseudoinverse matrices in matrix power series and matrix power products are obtained. A relationship between weighted pseudoinverse matrices the weighted normal pseudosolutions is established. Iterative methods for the calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions are constructed.     

Expansion of weighted pseudoinverse matrices with singular weights into matrix power products and iteration methods

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. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1269–1289, September, 2007.     

Expansion of Weighted Pseudoinverse Matrices in Matrix Power Products

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.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1539–1556, November, 2004.     

Weighted singular decomposition and weighted pseudoinversion of 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.     

Inversion of polynomial and rational matrices

The inversion of polynomial and rational matrices is considered. For regular matrices, three algorithms for computing the inverse matrix in a factored form are proposed. For singular matrices, algorithms of constructing pseudoinverse matrices are considered. The algorithms of inversion of rational matrices are based on the minimal factorization which reduces the problem to the inversion of polynomial matrices. A class of special polynomial matrices is regarded whose inverse matrices are also polynomial matrices. Inversion algorithms are applied to the solution of systems with polynomial and rational matrices. Bibliography: 3 titles. Translated by V. N. Kublanovskaya. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 97–109.     

Bifurcation of solutions of singular Fredholm boundary value problems

We obtain conditions for the bifurcation of solutions of linear singular Fredholm boundary value problems with a small parameter under the assumption that the unperturbed singular differential system can be reduced to central canonical form. Using the Vishik-Lyusternik method and the technique of Moore-Penrose pseudoinverse matrices, we suggest an algorithm for finding a family of linearly independent solutions of such boundary value problems for the general case in which the number of boundary conditions specified by a linear vector functional does not coincide with the number of unknowns in the singular differential system.     

On the Relative Gain Array (RGA) with singular and rectangular matrices

This paper identifies a significant deficiency in the literature on the application of the Relative Gain Array (RGA) formalism in the case of singular matrices. Specifically, it is shown that the conventional use of the Moore–Penrose pseudoinverse is inappropriate because it fails to preserve critical properties that can be assumed in the nonsingular case. It is then shown that such properties can be rigorously preserved using an alternative generalized matrix inverse.     

An approach to solving multiparameter algebraic problems

An approach to solving the following multiparameter algebraic problems is suggested: (1) spectral problems for singular matrices polynomially dependent on q≥2 spectral parameters, namely: the separation of the regular and singular parts of the spectrum, the computation of the discrete spectrum, and the construction of a basis that is free of a finite regular spectrum of the null-space of polynomial solutions of a multiparameter polynomial matrix; (2) the execution of certain operations over scalar and matrix multiparameter polynomials, including the computation of the GCD of a sequence of polynomials, the division of polynomials by their common divisor, and the computation of relative factorizations of polynomials; (3) the solution of systems of linear algebraic equations with multiparameter polynomial matrices and the construction of inverse and pseudoinverse matrices. This approach is based on the so-called ΔW-q factorizations of polynomial q-parameter matrices and extends the method for solving problems for one- and two-parameter polynomial matrices considered in [1–3] to an arbitrary q≥2. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 191–246. Translated by V. N. Kublanovskaya.     

A weighted pseudoinverse,generalized singular values,and constrained least squares problems

The weighted pseudoinverse providing the minimum semi-norm solution of the weighted linear least squares problem is studied. It is shown that it has properties analogous to those of the Moore-Penrose pseudoinverse. The relation between the weighted pseudoinverse and generalized singular values is explained. The weighted pseudoinverse theory is used to analyse least squares problems with linear and quadratic constraints. A numerical algorithm for the computation of the weighted pseudoinverse is briefly described.This work was supported in part by the Swedish Institute for Applied Mathematics.     

The least-squares method for matrices dependent on parameters

Some algorithms are suggested for constructing pseudoinverse matrices and for solving systems with rectangular matrices whose entries depend on a parameter in polynomial and rational ways. The cases of one- and two-parameter matrices are considered. The construction of pseudoinverse matrices are realized on the basis of rank factorization algorithms. In the case of matrices with polynomial occurrence of parameters, these algorithms are the ΔW-1 and ΔW-2 algorithms for one- and two-parameter matrices, respectively. In the case of matrices with rational occurrence of parameters, these algorithms are the irreducible factorization algorithms. This paper is a continuation of the author's studies of the solution of systems with one-parameter matrices and an extension of the results to the case of two-parameter matrices with polynomial and rational entries. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 176–185. This work was supported by the Russian Foundation of Fundamental Research (grant 94-01-00919). Translated by V. N. Kublanovskaya.     

A new method for computing Moore–Penrose inverse matrices

The Moore–Penrose inverse of an arbitrary matrix (including singular and rectangular) has many applications in statistics, prediction theory, control system analysis, curve fitting and numerical analysis. In this paper, an algorithm based on the conjugate Gram–Schmidt process and the Moore–Penrose inverse of partitioned matrices is proposed for computing the pseudoinverse of an m×n$m\times n$ real matrix A$A$ with m≥n$m\ge n$ and rank r≤n$r\le n$. Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that of pseudoinverses obtained by the other methods for large sparse matrices.     

ON STABLE PERTURBATIONS OF THE STIFFLY WEIGHTED PSEUDOINVERSE AND WEIGHTED LEAST SQUARES PROBLEM

In this paper we study perturbations of the stiffly weighted pseudoinverse （W^1/2 A）^＋W^1/2 and the related stiffly weighted least squares problem, where both the matrices A and W are given with W positive diagonal and severely stiff. We show that the perturbations to the stiffly weighted pseudoinverse and the related stiffly weighted least squares problem are stable, if and only if the perturbed matrices A = A ＋ δA satisfy several row rank preserving conditions.     

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.     

Upper bound and stability of scaled pseudoinverses

Summary. For given matrices and where is positive definite diagonal, a weighed pseudoinverse of is defined by and an oblique projection of is defined by . When is of full column rank, Stewart [3] and O'Leary [2] found sharp upper bound of oblique projections which is independent of , and an upper bound of weighed pseudoinverse by using the bound of . In this paper we discuss the sharp upper bound of over a set of positive diagonal matrices which does not depend on the upper bound of , and the stability of over . Received September 29, 1993 / Revised version received October 31, 1994     

Peakons,strings, and the finite Toda lattice

As is well‐known, the Toda lattice flow may be realized as an isospectral flow of a Jacobi matrix. A bijective map from a discrete string problem with positive weights to Jacobi matrices allows the pure peakon flow of the Camassa‐Holm equation to be realized as an isospectral Jacobi flow as well. This gives a unified picture of the Toda, Jacobi, and multipeakon flows, and leads to explicit solutions of the Jacobi flows via Stieltjes' determination of the continued fraction expansion of a Stieltjes transform. A simple modification produces a bijection from generalized strings, with positive and negative weights, to singular Jacobi matrices, and thus brings peakon/antipeakon flows into the same picture. © 2001 John Wiley & Sons, Inc.     

Lanczos Conjugate-Gradient Method and Pseudoinverse Computation on Indefinite and Singular Systems

This paper extends some theoretical properties of the conjugate gradient-type method FLR (Ref. 1) for iteratively solving indefinite linear systems of equations. The latter algorithm is a generalization of the conjugate gradient method by Hestenes and Stiefel (CG, Ref. 2). We develop a complete relationship between the FLR algorithm and the Lanczos process, in the case of indefinite and possibly singular matrices. Then, we develop simple theoretical results for the FLR algorithm in order to construct an approximation of the Moore-Penrose pseudoinverse of an indefinite matrix. Our approach supplies the theoretical framework for applications within unconstrained optimization. This work was partially supported by the MIUR, FIRB Research Program on Large-Scale Nonlinear Optimization and by the Ministero delle Infrastrutture e dei Trasporti in the framework of the Research Program on Safety. The author thanks Stefano Lucidi and Massimo Roma for fruitful discussions plus the Associate Editor for effective comments.     

Nonnegative matrices having nonnegative Drazin pseudoinverses

Necessary and sufficient conditions for nonnegative matrices having nonnegative Drazin pseudoinverses are obtained. A decomposition theorem which characterizes the class of all nonnegative matrices with nonnegative Drazin pseudoinverses is proved, thus answering a question raised by several people. It is also shown that if a row (or column) stochastic matrix has a nonnegative Drazin pseudoinverse A(^d), then A(^d) is some power of A. These results extend known results for nonnegative group-monotone matrices.     

The generalized Cauchy problem for singularly perturbed impulsive systems in the critical case

We construct an asymptotic expansion for solutions to nonlinear singularly perturbed systems of impulsive differential equations. We successively determine all terms of the asymptotic expansion by means of pseudoinverse matrices and orthoprojections.     

Hyperbolic forms associated with cyclic weighted shift matrices

The singular points of the curve of a hyperbolic form associated with a cyclic weighted shift matrix are examined. It is shown that the singular points of such a curve are real nodes. Some results related the numerical ranges of cyclic weighted shift matrices are presented. In particular, the existence of flat portions on the boundary of the numerical range depends on the reducibility of the hyperbolic form. Further, an algebraic method is provided for the decomposition of reducible form which leads to a criterion for the periodicity of the weights.     

带ARMA有色噪声的广义系统信息融合Kalman平滑器

对于带自回归滑动平均(ARMA)有色观测噪声的多传感器广义离散随机线性系统,应用奇异值分解,提出了广义系统多传感器信息融合状态平滑问题。基于Kalman滤波方法,在线性最小方差信息融合准则下,给出了按矩阵加权融合降阶稳态广义Kalman平滑器。为了计算最优加权,提出了局部平滑误差协方差阵的计算公式。一个Monte Carlo仿真例子说明了所提方法的有效性。