Projection matrices, generalized inverse matrices, and singular value decomposition.
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.
scientific article; zbMATH DE number 5882803