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A Matrix Handbook for Statisticians 矩阵与统计.pdf

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    A Matrix Handbook for Statisticians
    CONTENTS
    Preface
    1. Notation
    2. Vectors, Vector Spaces, and Convexity
    3. Rank
    4. Matrix Functions: Inverse, Transpose, Trace, Determinant, and Norm
    5. Complex, Hermitian, and Related Matrices
    6. Eigenvalues, Eigenvectors, and Singular Values
    7. Generalized Inverses
    8. Some Special Matrices
    9. Non-Negative Vectors and Matrices
    10. Positive Definite and Non-negative Definite Matrices
    11. Special Products and Operators
    12. Inequalities
    13. Linear Equations
    14. Partitioned Matrices
    15. Patterned Matrices
    16. Factorization of Matrices
    17. Differentiation
    18. Jacobians
    19. Matrix Limits, Sequences, and Series
    20. Random Vectors
    21. Random Matrices
    22. Inequalities for Probabilities and Random Variables
    23. Majorization
    24. Optimization and Matrix Approximation
    References
    Index
    Eagle Hill
    A MATRIX HANDBOOK FOR STATISTICIANS George A. F. Seber Department of Statistics University of Auckland Auckland, New Zealand B I C E N T E N N I A L B I C E N T E N N I A L WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication
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    A MATRIX HANDBOOK FOR STATISTICIANS
    THE WlLEY BICENTENNIAL-KNOWLEDGE F O R GENERATIONS G a c h generation has its unique needs and aspirations. When Charles Wiley first opened his small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity. And we were there, helping to define a new American literary tradition. Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused on building the future. Once again, we were there, supplying the critical scientific, technical, and engineering knowledge that helped frame the world. Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own borders and a new international community was born. Wiley was there, expanding its operations around the world to enable a global exchange of ideas, opinions, and know-how. For 200 years, Wiley has been an integral part of each generation's journey, enabling the flow of information and understanding necessary to meet their needs and fulfill their aspirations. Today, bold new technologies are changing the way we live and learn. Wiley will be there, providing you the must-have knowledge you need to imagine new worlds, new possibilities, and new opportunities. Generations come and go, but you can always count on Wiley to provide you the knowledge you need, when and where you need it! n WILLIAM J. PESCE PRESIDENT AND CHIEF ExmzunvE OFFICER PETER BOOTH WILEY CHAIRMAN OF THE BOARD
    A MATRIX HANDBOOK FOR STATISTICIANS George A. F. Seber Department of Statistics University of Auckland Auckland, New Zealand B I C E N T E N N I A L B I C E N T E N N I A L WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication
    Copyright 0 2008 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at m.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 11 1 River Street, Hoboken, NJ 07030, (201) 748-601 1, fax (201) 748-6008, or online at http://www.wiley.comgo/permission. Limit of LiabilitylDisclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information about Wiley products, visit our web site at www.wiley.com. Wiley Bicentennial Logo: Richard J. Pacific0 Library of Congress Cataloging-in-Publieation Data: Seber, G. A. F. (George Arthur Frederick), 1938- A matrix handbook for statisticians I George A.F. Seber. p.; cm. Includes bibliographical references and index. ISBN 978-0-471-74869-4 (cloth ) 1. Matrices. 2. Statistics. I. Title. QA188.S43 2007 5 1 2 . 9 ' 4 3 4 6 ~ 2 2 2007024691 Printed in the United States of America. 1 0 9 8 7 6 5 4 3 2 1
    CONTENTS Preface 1 Notation 1.1 General Definitions 1.2 1.3 Glossary of Notation Some Continuous Univariate Distributions 2 Vectors, Vector Spaces, and Convexity 2.1 2.2 2.3 Sums and Intersections of Subspaces Span and Basis Isomorphism Vector Spaces 2.1.1 Definitions 2.1.2 Quadratic Subspaces 2.1.3 2.1.4 2.1.5 Inner Products 2.2.1 Definition and Properties 2.2.2 2.2.3 Orthogonality 2.2.4 Column and Null Spaces Projections Functionals xvi 7 7 7 9 10 11 12 13 13 15 16 18 20 V
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