Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition
Aside from distribution theory, projections and the singular value decomposition (SVD) are the two most important concepts for understanding the basic mechanism of multivariate analysis. The former underlies the least squares estimation in regression anal
- PDF / 3,976,677 Bytes
- 244 Pages / 483.766 x 730.083 pts Page_size
- 84 Downloads / 219 Views
Haruo Yanai Kei Takeuchi Yoshio Takane
Statistics for Social and Behavioral Sciences Advisors: S.E. Fienberg W.J. van der Linden
For other titles published in this series, go to http://www.springer.com/series/3463
Haruo Yanai • Kei Takeuchi • Yoshio Takane
Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition
Haruo Yanai Department of Statistics St. Luke’s College of Nursing 10-1 Akashi-cho Chuo-ku Tokyo 104-0044 Japan [email protected]
Kei Takeuchi 2-34-4 Terabun Kamakurashi Kanagawa-ken 247-0064 Japan [email protected]
Yoshio Takane Department of Psychology McGill University 1205 Dr. Penfield Avenue Montreal Québec H3A 1B1 Canada [email protected]
ISBN 978-1-4419-9886-6 e-ISBN 978-1-4419-9887-3 DOI 10.1007/978-1-4419-9887-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011925655 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface All three authors of the present book have long-standing experience in teaching graduate courses in multivariate analysis (MVA). These experiences have taught us that aside from distribution theory, projections and the singular value decomposition (SVD) are the two most important concepts for understanding the basic mechanism of MVA. The former underlies the least squares (LS) estimation in regression analysis, which is essentially a projection of one subspace onto another, and the latter underlies principal component analysis (PCA), which seeks to find a subspace that captures the largest variability in the original space. Other techniques may be considered some combination of the two. This book is about projections and SVD. A thorough discussion of generalized inverse (g-inverse) matrices is also given because it is closely related to the former. The book provides systematic and in-depth accounts of these concepts from a unified viewpoint of linear transformations in finite dimensional vector spaces. More specifically, it shows that projection matrices (projectors) and g-inverse matrices can be defined in various ways so that a vector space is decomposed into a direct-sum of (disjoint) subspaces. This book gives analogous decompositions of matrices and discusses their possible appl
Data Loading...
