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2018 | Buch

Generalized Inverses: Theory and Computations

verfasst von: Guorong Wang, Prof. Yimin Wei, Sanzheng Qiao

Verlag: Springer Singapore

Buchreihe : Developments in Mathematics

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Über dieses Buch

This book begins with the fundamentals of the generalized inverses, then moves to more advanced topics.

It presents a theoretical study of the generalization of Cramer's rule, determinant representations of the generalized inverses, reverse order law of the generalized inverses of a matrix product, structures of the generalized inverses of structured matrices, parallel computation of the generalized inverses, perturbation analysis of the generalized inverses, an algorithmic study of the computational methods for the full-rank factorization of a generalized inverse, generalized singular value decomposition, imbedding method, finite method, generalized inverses of polynomial matrices, and generalized inverses of linear operators. This book is intended for researchers, postdocs, and graduate students in the area of the generalized inverses with an undergraduate-level understanding of linear algebra.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Equation Solving Generalized Inverses
Abstract
There are various ways to introduce the generalized inverses. We introduce them by considering the problem of solving systems of linear equations.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 2. Drazin Inverse
Abstract
In Chap. 1, we discussed the Moore-Penrose inverse and the \(\{i, j, k\}\) inverses which possess some “inverse-like” properties. The \(\{ i, j, k \}\) inverses provide some types of solution, or the least-square solution, for a system of linear equations just as the regular inverse provides a unique solution for a nonsingular system of linear equations. Hence the \(\{ i, j, k \}\) inverses are called equation solving inverses. However, there are some properties of the regular inverse matrix that the \(\{ i, j, k \}\) inverses do not possess.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 3. Generalization of the Cramer’s Rule and the Minors of the Generalized Inverses
Abstract
It is well known that the Cramer’s rule for the solution \(\mathbf {x}\) of a nonsingular equation
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 4. Reverse Order and Forward Order Laws for
Abstract
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.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 5. Computational Aspects
Abstract
It follows from Chap. 1 that the six important kinds of generalized inverse: the M-P inverse \(A^\dag \), the weighted M-P inverse \(A_{MN}^{\dag }\), the group inverse \(A_g\), the Drazin inverse \(A_d\), the Bott-Duffin inverse \(A_{(L)}^{(-1)}\) and the generalized Bott-Duffin inverse \(A_{(L)}^{(\dag )}\) are all the generalized inverse \(A_{T,S}^{(2)}\), which is the \(\{ 2 \}\)-inverse of A with the prescribed range T and null space S.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 6. Structured Matrices and Their Generalized Inverses
Abstract
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.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 7. Parallel Algorithms for Computing the Generalized Inverses
Abstract
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.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 8. Perturbation Analysis of the Moore-Penrose Inverse and the Weighted Moore-Penrose Inverse
Abstract
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 \(B=A+E\) of A. It is natural to ask if the generalized inverse of B is close to that of A when the perturbation E is sufficiently small.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 9. Perturbation Analysis of the Drazin Inverse and the Group Inverse
Abstract
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.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 10. Generalized Inverses of Polynomial Matrices
Abstract
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, \((\mathbb {R}[x])^{m \times n}\) and \((\mathbb {R}^{m \times n}) [x]\) are isomorphic. In other words, extending the entries of matrices to polynomials is the same as extending the coefficients of polynomials to matrices.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 11. Moore-Penrose Inverse of Linear Operators
Abstract
Before Moore introduced the generalized inverse of matrices by algebraic methods, Fredholm, Hilbert, Schmidt, Bounitzky, Hurwitz and other mathematicians had studied the generalized inverses of integral operators and differential operators. Recently, due to the development of science and technology and the need for practical problems, researchers are very interested in the study of the generalized inverses of linear operators in abstract spaces.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Chapter 12. Operator Drazin Inverse
Abstract
Let X be a Hilbert space and L(X) be the vector space of the linear operators from X into X. We denote the set of bounded linear operators from X into X by B(X). In this chapter, we will investigate the definition, basic properties, representation theorem and computational methods for the Drazin inverse of an operator \(T \in B(X)\), \(\mathcal {R}(T^k)\) is closed, where \(k=\mathrm {Ind}(T)\) is the index of T.
Guorong Wang, Yimin Wei, Sanzheng Qiao
Backmatter
Metadaten
Titel
Generalized Inverses: Theory and Computations
verfasst von
Guorong Wang
Prof. Yimin Wei
Sanzheng Qiao
Copyright-Jahr
2018
Verlag
Springer Singapore
Electronic ISBN
978-981-13-0146-9
Print ISBN
978-981-13-0145-2
DOI
https://doi.org/10.1007/978-981-13-0146-9