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Generalized Inverses: Theory and Computations

Part of the Developments in Mathematics Book Series (volume 53)

DOI: 10.1007/978-981-13-0146-9

ISBN: 978-981-13-0145-2

Publisher: Springer Singapore - Springer Journals

May 13, 2018

1 - 10 of 12 Chapters

LitStream

Chapter 1Equation Solving Generalized Inverses

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 1 –64

[There are various ways to introduce the generalized inverses. We introduce them by considering the problem of solving systems of linear equations.]
LitStream

Chapter 2Drazin Inverse

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 65 –90

[In Chap. 1, we discussed the Moore-Penrose inverse and the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

Chapter 3Generalization of the Cramer’s Rule and the Minors of the Generalized Inverses

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 91 –151

[It is well known that the Cramer’sCramer’s rule rule for the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

Chapter 4Reverse Order and Forward Order Laws for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{T,S}^{(2)}$$\end{document}

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 153 –174

[The reverse order law for the generalized inverses of a matrix product yields a class of interesting fundamental problems in the theory of the generalized inverses of matrices. They have attracted considerable attention since the middle 1960s.]
LitStream

Chapter 5Computational Aspects

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 175 –224

[It follows from Chap. 1 that the six important kinds of generalized inverse: the M-P inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}...
LitStream

Chapter 6Structured Matrices and Their Generalized Inverses

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 225 –231

[A matrix is considered structured if its structure can be exploited to obtain efficient algorithms. Examples of structured matrices include Toeplitz, Hankel, circulant, Vandermonde, Cauchy, sparse. A matrix is called Toeplitz if its entries on the same diagonal are equal.]
LitStream

Chapter 7Parallel Algorithms for Computing the Generalized Inverses

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 233 –261

[The UNIVersal Automatic Computer (UNIVAC I) and the machines built in 1940s and mid 1950s are often referred to as the first generation of computers.]
LitStream

Chapter 8Perturbation Analysis of the Moore-Penrose Inverse and the Weighted Moore-Penrose Inverse

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 263 –289

[Let A be a given matrix. When computing a generalized inverse of A, due to rounding error, we actually obtain the generalized inverse of a perturbed matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}...
LitStream

Chapter 9Perturbation Analysis of the Drazin Inverse and the Group Inverse

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 291 –306

[Having studied the perturbation of the M-P inverse and the weighted M-P inverse, we now turn to the perturbation analysis of the Drazin and group inverses.]
LitStream

Chapter 10Generalized Inverses of Polynomial Matrices

Wang, Guorong; Wei, Yimin; Qiao, Sanzheng

Pages: 307 –316

[A polynomial matrix is a matrix whose entries are polynomials. Equivalently, a polynomial matrix can be expressed as a polynomial with matrix coefficients. Formally speaking, in the univariable case, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}...

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