Projection matrices, generalized inverse matrices, and singular value decomposition.
type: Item
instance of: scholarly article
title: Projection matrices, generalized inverse matrices, and singular value decomposition.
zbMATH Open document ID: 1279.15003
DOI: 10.1007/978-1-4419-9887-3
author: Haruo Yanai
author: Kei Takeuchi
author: Yoshio Takane
published in: Statistics for Social and Behavioral Sciences
publication date: 2011-05-02T00:00:00Z
review text: The book under review is devoted, mainly, to projections and singular value decomposition (SVD). The text consists of six chapters and starts with Chapter 1 (Fundamentals of linear algebra) that establishes notations and introduces preliminaries from linear algebra. Chapter 2 (Projection matrices) plays an important role and is devoted to oblique projectors whereas the more commonly used orthogonal projectors are special cases of them. A particular attention is given to decompositions of projectors and corresponding subspaces. These geometrical considerations play an important role in Chapter 3 (General inverse matrices) and in Chapter 4 (Explicit representations). Chapter 5 (Singular value decomposition (SVD)) deals with connections between SVD, projectors and general inverse matrices. In the concluding Chapter 6 (Various applications) the authors demonstrate that the concepts given in the preceding chapters play important roles in applied fields such as linear regression analysis (the method of least squares and multiple regression analysis; multiple correlation coefficients and their partitions; the Gauss-Markov model), analysis of variance (one-way, two-way and three-way design; Cochran's theorem), multivariate analysis (canonical correlation and canonical determination analysis; principal component analysis; distance and projection matrices), linear simultaneous equations (QR decomposition by the Gram-Schmidt orthogonalization method; QR decomposition by the Householder transformation; decomposition by projectors). 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.
reviewed by: Edward L. Pekarev
Mathematics Subject Classification ID: 15-01
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 00A06
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 62J10
Mathematics Subject Classification ID: 62H20
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62H35
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F25
zbMATH DE Number: 5882803
zbMATH Keywords: matrix analysis
zbMATH Keywords: eigenvalues
zbMATH Keywords: singular values
zbMATH Keywords: matrix factorization
zbMATH Keywords: singular value decomposition
zbMATH Keywords: least squares solution
zbMATH Keywords: generalized inverse
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: QR decomposition
zbMATH Keywords: canonical correlation analysis
zbMATH Keywords: canonical discriminant analysis
zbMATH Keywords: analysis of variance
zbMATH Keywords: exercises
zbMATH Keywords: textbook
zbMATH Keywords: linear regression
zbMATH Keywords: multiple regression
zbMATH Keywords: Gauss-Markov model
zbMATH Keywords: principal component analysis
zbMATH Keywords: Gram-Schmidt orthogonalization
zbMATH Keywords: Householder transformation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4419-9887-3
OpenAlex ID: W2478027691
description: scientific article; zbMATH DE number 5882803