Matrix analysis for statistics.

*(English)*Zbl 0872.15002
Wiley Series in Probability and Mathematical Statistics. New York, NY: Wiley. xii, 426 p. (1997).

This book was written with an entire graduate course devoted to matrix methods useful in statistics (e.g. regression analysis, multivariate analysis, linear models, stochastic processes) in mind. The author believes that with selective omission of some sections, all nine chapters of this book could be covered in a one-semester course.

The chapter headings are as follows. 1. A review of elementary matrix algebra. 2. Vector spaces. 3. Eigenvalues and eigenvectors. 4. Matrix factorizations and matrix norms. 5. Generalized inverses. 6. Systems of linear equations. 7. Special matrices and matrix operators. 8. Matrix derivatives and related topics. 9. Some special topics related to quadratic forms.

There are 3 pages of references, including many books, pertinent to the subject matter generally or to specific topics. There is a subject-matter index (pp. 421-426) and problem sets for the chapters.

The reviewer’s impression is that the book is rather ambitious for the audience at which it is aimed, and is, rather, a quite substantial treatment of useful topics of matrix theory per se, including, in most cases, proofs, and deserving just the name Matrix analysis. In particular some of the topics (e.g. circulant and Toeplitz matrices, Hadamard and Vandermonde matrices) seem to be less needed by the intended readership; as is the topic of non-negative matrices developed “along the lines of the derivations, based on matrix norms, given by R. A. Horn and C. R. Johnson [Matrix analysis, Cambridge Univ. Press. XIII, 561 p. (1985; Zbl 0576.15001)]” (that is: their Matrix analysis), presumably in connection with finite Markov chains. Nevertheless, the reviewer is pleased to have this book as a reference.

The chapter headings are as follows. 1. A review of elementary matrix algebra. 2. Vector spaces. 3. Eigenvalues and eigenvectors. 4. Matrix factorizations and matrix norms. 5. Generalized inverses. 6. Systems of linear equations. 7. Special matrices and matrix operators. 8. Matrix derivatives and related topics. 9. Some special topics related to quadratic forms.

There are 3 pages of references, including many books, pertinent to the subject matter generally or to specific topics. There is a subject-matter index (pp. 421-426) and problem sets for the chapters.

The reviewer’s impression is that the book is rather ambitious for the audience at which it is aimed, and is, rather, a quite substantial treatment of useful topics of matrix theory per se, including, in most cases, proofs, and deserving just the name Matrix analysis. In particular some of the topics (e.g. circulant and Toeplitz matrices, Hadamard and Vandermonde matrices) seem to be less needed by the intended readership; as is the topic of non-negative matrices developed “along the lines of the derivations, based on matrix norms, given by R. A. Horn and C. R. Johnson [Matrix analysis, Cambridge Univ. Press. XIII, 561 p. (1985; Zbl 0576.15001)]” (that is: their Matrix analysis), presumably in connection with finite Markov chains. Nevertheless, the reviewer is pleased to have this book as a reference.

Reviewer: Eugene Seneta (Sydney)

##### MSC:

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

15A09 | Theory of matrix inversion and generalized inverses |

15A03 | Vector spaces, linear dependence, rank, lineability |

15A06 | Linear equations (linear algebraic aspects) |

15A18 | Eigenvalues, singular values, and eigenvectors |

15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |

26B12 | Calculus of vector functions |

15A15 | Determinants, permanents, traces, other special matrix functions |

15A04 | Linear transformations, semilinear transformations |

15B36 | Matrices of integers |

15A23 | Factorization of matrices |

15A21 | Canonical forms, reductions, classification |

62J05 | Linear regression; mixed models |