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

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.

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“…For the other definitions of weighted pseudoinverse matrices with singular weights and the results of their investigation, see [10][11][12].…”

Section: Notation Definitions Known Facts and Auxiliary Statementsmentioning

confidence: 99%

Necessary and Sufficient Conditions for the Existence of Weighted Singular-Valued Decompositions of Matrices with Singular Weights

2015

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“…The study of the properties of weighted pseudoinverse matrices and weighted normal pseudosolutions, as well as the construction of methods for solving these and other problems, are devoted to the works of Mitra, Rao, Van Loan, Wang, Galba, Deineka, Sergienko, Ben-Israel, Elden, Wei, Wei, Ward etc. Weighted pseudoinverse matrices and weighted normal pseudosolutions with degenerate weights were studied in [3][4][5]. The existence and uniqueness of weighted pseudoinverse matrices with indefinite and mixed weights, as well as some of their properties, were described in [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Weighted Least Squares Perturbation Theory

Химич¹,

Nikolaevskaya²,

Baranov³

2023

Matrix Theory - Classics and Advances

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The interest in the problem of weighted pseudoinverse matrices and the problem of weighted least squares (WLS) is largely due to their numerous applications. In particular, the problem of WLS is used in the design and optimization of building structures, in tomography, in statistics, etc. The first part of the chapter is devoted to the sensitivity of the solution to the WLS problem with approximate initial data. The second part investigates the properties of a SLAE with approximate initial data and presents an algorithm for finding a weighted normal pseudo solution of a WLS problem with approximate initial data, an algorithm for solving a WLS problem with symmetric positive semidefinite matrices and an approximate right side and also a parallel algorithm for solving a WLS problem. The third part is devoted to the analysis of the reliability of computer solutions of the WLS problem with approximate initial data. Here, estimates of the total error of the WLS problem are presented, and also software-algorithmic approaches to improving the accuracy of computer solutions.

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Mathematical Modeling and Analysis of Complex Processes

Сергиенко

2014

Topical Directions of Informatics

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

Cited by 14 publications

References 10 publications

Necessary and Sufficient Conditions for the Existence of Weighted Singular-Valued Decompositions of Matrices with Singular Weights

Necessary and Sufficient Conditions for the Existence of Weighted Singular-Valued Decompositions of Matrices with Singular Weights

Weighted Least Squares Perturbation Theory

Mathematical Modeling and Analysis of Complex Processes

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