Nonnegative matrices and applications.

*(English)*Zbl 0879.15015
Encyclopedia of Mathematics and Its Applications. 64. Cambridge: Cambridge University Press. xiii, 336 p. (1997).

The claims set out at the beginning include: “This book presents an integrated treatment of the theory of nonnegative matrices, emphasizing connections with the themes of game theory, combinatorics, inequalities, optimization, and mathematical economics…The book begins with the basics of the subject, such as the Perron-Frobenius theorem.…Each of the later chapters is devoted to an area of applications…These applications have been carefully chosen both for their elegant mathematical content and for their accessibility…About half of the material in the book presents standard topics in a novel form, the remaining portion reports many new results in matrix theory for the first time in book form.”

The chapter headings are as follows: 1. Perron-Frobenius theory and matrix games (this includes one section, 1.9, on finite Markov chains). 2. Doubly stochastic matrices. 3. Inequalities. 4. Conditionally positive definite matrices. 5. Topics in combinatorial theory. 6. Scaling problems and their applications. 7. Special matrices in economic models.

Each chapter is concluded by exercises. The list of references is followed by an Index and an Author Index.

This book is one of several to have now appeared dedicated to the topic of nonnegative matrices in whole or in part. The authors list, but mention only in passing, the books of the reviewer [Nonnegative matrices, An introduction to theory and applications (1973; Zbl 0278.15011)], H. Minc [Nonnegative matrices (1988; Zbl 0638.15008)] and A. Berman and R. J. Plemmons [Nonnegative matrices in the mathematical sciences (1994; Zbl 0815.15016)]. They seem to be unaware that there is a second edition of the reviewer’s book with modified title and contents [Non-negative matrices and Markov chains (1981; Zbl 0471.60001)] which contains a treatment of several topics covered in their own (e.g. their §6.1 Hilbert’s projective metric; and some material on scaling).

The present book can be seen to treat a collection of diverse topics, and could have been titled more appropriately. The authors very much focus on the mathematical derivations rather than ‘applications’, and this reviewer missed clearly expressed motivations for the sequence of topics as treated in the individual chapters. The reference list is extensive, but the contents of the book tend to be based on the authors’ own interests and contributions. The first named author has 31 entries (solo and joint) in the list of references, beginning with his 1981 Ph.D. thesis at the University of Illinois.

The chapter headings are as follows: 1. Perron-Frobenius theory and matrix games (this includes one section, 1.9, on finite Markov chains). 2. Doubly stochastic matrices. 3. Inequalities. 4. Conditionally positive definite matrices. 5. Topics in combinatorial theory. 6. Scaling problems and their applications. 7. Special matrices in economic models.

Each chapter is concluded by exercises. The list of references is followed by an Index and an Author Index.

This book is one of several to have now appeared dedicated to the topic of nonnegative matrices in whole or in part. The authors list, but mention only in passing, the books of the reviewer [Nonnegative matrices, An introduction to theory and applications (1973; Zbl 0278.15011)], H. Minc [Nonnegative matrices (1988; Zbl 0638.15008)] and A. Berman and R. J. Plemmons [Nonnegative matrices in the mathematical sciences (1994; Zbl 0815.15016)]. They seem to be unaware that there is a second edition of the reviewer’s book with modified title and contents [Non-negative matrices and Markov chains (1981; Zbl 0471.60001)] which contains a treatment of several topics covered in their own (e.g. their §6.1 Hilbert’s projective metric; and some material on scaling).

The present book can be seen to treat a collection of diverse topics, and could have been titled more appropriately. The authors very much focus on the mathematical derivations rather than ‘applications’, and this reviewer missed clearly expressed motivations for the sequence of topics as treated in the individual chapters. The reference list is extensive, but the contents of the book tend to be based on the authors’ own interests and contributions. The first named author has 31 entries (solo and joint) in the list of references, beginning with his 1981 Ph.D. thesis at the University of Illinois.

Reviewer: Eugene Seneta (Sydney)

##### MSC:

15B48 | Positive matrices and their generalizations; cones of matrices |

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

15A45 | Miscellaneous inequalities involving matrices |

05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |

91A20 | Multistage and repeated games |

15B51 | Stochastic matrices |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

15A12 | Conditioning of matrices |

91B60 | Trade models |