Generalized Inverses : Theory and Applications

Thomas N. E. Greville

ISBN 10: 1441918140 / ISBN 13: 9781441918147

Published by Springer US, Springer New York, 2010

About this Item

Druck auf Anfrage Neuware - Printed after ordering - 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.

Seller Inventory # 9781441918147

Bibliographic Details

• Title: Generalized Inverses : Theory and Applications
• Publisher: Springer US, Springer New York
• Publication Date: 2010
• Language: English
• ISBN 10: 1441918140
• ISBN 13: 9781441918147
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 235W x 155H x 24D millimeters 235 millimeters width by 155 millimeters height by 24 millimeters depth
• Weight: 663 grams

About this title

Synopsis

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.

About the Author

 

Adi Ben-Israel is Professor of Operations Research, Business and Mathematics at Rutgers University, New Brunswick, NJ. Previously he was Professor of Applied Mathematics at the University of Delaware, Northwestern University, and the Technion-Israel Institute of Technology.

The late Thomas N.E. Greville was Professor of Mathematics, and a member of the US Army Mathematics Research Center at the University of Wisconsin, Madison, WI.