Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition

  • Haruo Yanai
  • Kei Takeuchi
  • Yoshio Takane

Part of the Statistics for Social and Behavioral Sciences book series (SSBS)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Haruo Yanai, Kei Takeuchi, Yoshio Takane
    Pages 1-24
  3. Haruo Yanai, Kei Takeuchi, Yoshio Takane
    Pages 25-54
  4. Haruo Yanai, Kei Takeuchi, Yoshio Takane
    Pages 55-86
  5. Haruo Yanai, Kei Takeuchi, Yoshio Takane
    Pages 87-123
  6. Haruo Yanai, Kei Takeuchi, Yoshio Takane
    Pages 125-149
  7. Haruo Yanai, Kei Takeuchi, Yoshio Takane
    Pages 151-203
  8. Haruo Yanai, Kei Takeuchi, Yoshio Takane
    Pages 205-228
  9. Haruo Yanai, Kei Takeuchi, Yoshio Takane
    Pages 229-232
  10. Back Matter
    Pages 233-234

About this book

Introduction

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 analysis, which is essentially a projection of one subspace onto another, and the latter underlies principal component analysis, which seeks to find a subspace that captures the largest variability in the original space.

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 finite dimensional vector spaces. More specially, 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. Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition will be useful for researchers, practitioners, and students in applied mathematics, statistics, engineering, behaviormetrics, and other fields.

Keywords

Multivariate analysis g-inverse matrices linear transformations projections singular value decomposition

Authors and affiliations

  • Haruo Yanai
    • 1
  • Kei Takeuchi
    • 2
  • Yoshio Takane
    • 3
  1. 1., Department of StatisticsSt. Luke’s College of NursingTokyoJapan
  2. 2.Kanagawa-kenJapan
  3. 3., Department of PsychologyMcGill UniversityMontrealCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-9887-3
  • Copyright Information Springer Science+Business Media, LLC 2011
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4419-9886-6
  • Online ISBN 978-1-4419-9887-3
  • About this book