Generalized inverses : theory and applications / Adi Ben-Israel, Thomas N.E. Grenville.

By:

Ben-Israel, Adi

Contributor(s):

Greville, T. N. E. (Thomas Nall Eden), 1910-1998

Material type: Text

TextSeries: CMS books in mathematics ; 15Publication details: New York : Springer, c2003.Edition: 2nd edDescription: xv, 420 p. ; 25 cmISBN:

9781441918147

Subject(s):

Matrix inversion

DDC classification:

512.943 21 BEN

LOC classification:

QA188 .B46 2003

Online resources:

Contents:

Introduction Pages 1-5 Preliminaries Pages 6-39 Existence and Construction of Generalized Inverses Pages 40-51 Linear Systems and Characterization of Generalized Inverses Pages 52-103 Minimal Properties of Generalized Inverses Pages 104-151 Spectral Generalized Inverses Pages 152-174 Generalized Inverses of Partitioned Matrices Pages 175-200 A Spectral Theory for Rectangular Matrices Pages 201-256 Computational Aspects of Generalized Inverses Pages 257-281 Miscellaneous Applications Pages 282-329 Generalized Inverses of Linear Operators between Hilbert Spaces Pages 330-369 Back Matter Pages 370-420

Summary: 1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse, ?1 denoted by A , such that ?1 ?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus, ?1 ?1 (A ) = A, T ?1 ?1 T (A ) =(A ) , ? ?1 ?1 ? (A ) =(A ) , ?1 ?1 ?1 (AB) = B A , T ? where A and A , respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number ? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to ?,if Ax = ?x. ?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.

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Holdings
Item type	Current library	Collection	Call number	Status	Date due	Barcode
Project book	CUTN Central Library Sciences	Non-fiction	512.943 BEN (Browse shelf(Opens below))	Checked out to Renuka Devi V (20019T)	31/01/2024	48833

Includes bibliographical references and index.

Introduction
Pages 1-5
Preliminaries
Pages 6-39
Existence and Construction of Generalized Inverses
Pages 40-51
Linear Systems and Characterization of Generalized Inverses
Pages 52-103
Minimal Properties of Generalized Inverses
Pages 104-151
Spectral Generalized Inverses
Pages 152-174
Generalized Inverses of Partitioned Matrices
Pages 175-200
A Spectral Theory for Rectangular Matrices
Pages 201-256
Computational Aspects of Generalized Inverses
Pages 257-281
Miscellaneous Applications
Pages 282-329
Generalized Inverses of Linear Operators between Hilbert Spaces
Pages 330-369
Back Matter
Pages 370-420

1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse, ?1 denoted by A , such that ?1 ?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus, ?1 ?1 (A ) = A, T ?1 ?1 T (A ) =(A ) , ? ?1 ?1 ? (A ) =(A ) , ?1 ?1 ?1 (AB) = B A , T ? where A and A , respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number ? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to ?,if Ax = ?x. ?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.

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