Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition

By: Haruo Yanai, Kei Takeuchi, Yoshio Takane

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Paperback | 28 May 2013

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248 Pages

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23.39 x 15.6 x 1.32

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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.

Industry Reviews

From the reviews:

"The book under review is devoted, mainly, to projections and singular value decomposition (SVD). ... Each chapter has some exercises. Many examples illustrate the presented material very well. The book should serve as a useful reference on projectors, general inverses and SVD, it is of interest to those working in matrix analysis, it can be recommended for graduate students as well as for professionals." (Edward L. Pekarev, zbMATH, Vol. 1279, 2014)

"Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition is more suitable for readers who enjoy mathematics for its beauty. ... this book has been prepared with great care. It was meant to serve as a useful reference on projectors 'for researchers, practitioners and students in applied mathematics, engineering, and behaviormetrics'. I expect it to succeed in this respect." (Jos M. F. ten Berge, Psychometrika, Vol. 77 (3), July, 2012)

"This book is devoted to projectors (projection matrices) and singular value decomposition (SVD). A complete discussion of the closely related topic of generalized inverses (g-inverses) is provided. ... should be of interest and serve as a reference to researchers and students in applied mathematics, statistics, engineering, and other related fields. The central properties of projections and singular value decomposition are presented in full detail and an excellent bibliography is provided." (Ronald L. Smith, Mathematical Reviews, Issue 2012 c)

"Researchers, practitioners, and students in applied mathematics, statistics, engineering, behaviour metrics, and other fields. ... this book is a very useful collection of very important matrix results related to statistical multivariate analysis. ... The authors earn congratulations for careful and clear writing, nice-looking format, and especially for numerous figures that illustrate the geometry of the concepts. Moreover, the exercises with their solutions are warmly welcome." (Simo Puntanen, International Statistical Review, Vol. 79 (3), 2011)

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