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**Nonnegative matrices.**
*(English)*
Zbl 0638.15008

Wiley-Interscience Series in Discrete Mathematics and Optimization. New York etc.: Wiley. xii, 206 p. £35.90 (1988).

Matrices with nonnegative entries (=nonnegative matrices) are studied in this book. The author is well known as coauthor of the joint book with M. Marcus [A survey of matrix theory and matrix inequalities (1964; Zbl 0126.024)] in which nonnegative matrices are treated in §II.5.

According to the author, his purpose is a) to provide a text at the undergraduate or graduate levels, and b) to write a self-contained reference work for mathematicians and scientists. The book consists of 7 chapters. The first 3 chapters contain a modern presentation of the Perron-Frobenius theory. Chapters 4 to 7 are independent of each other but they depend on the first three chapters. Ch. 4 deals with combinatorial properties of nonnegative matrices (the Frobenius-König theorem, connections with graph theory, decomposable and reducible matrices, and bounds for permanents are discussed). Ch. 5 deals with doubly stochastic matrices (where, among other things, the proof of the van der Waerden conjecture is given). Ch. 6 deals with various classes of nonnegative matrices which arise in applications (stochastic, oscillatory, totally nonnegative and \(M\)-matrices). Ch. 7 deals with inverse eigenvalue problems, inverse spectrum problems and with similarity to nonnegative and to double stochastic matrices. At the end of each chapter the reader finds problems and references.

The topic of the book is important and interesting and the book can be used by a broad audience.

According to the author, his purpose is a) to provide a text at the undergraduate or graduate levels, and b) to write a self-contained reference work for mathematicians and scientists. The book consists of 7 chapters. The first 3 chapters contain a modern presentation of the Perron-Frobenius theory. Chapters 4 to 7 are independent of each other but they depend on the first three chapters. Ch. 4 deals with combinatorial properties of nonnegative matrices (the Frobenius-König theorem, connections with graph theory, decomposable and reducible matrices, and bounds for permanents are discussed). Ch. 5 deals with doubly stochastic matrices (where, among other things, the proof of the van der Waerden conjecture is given). Ch. 6 deals with various classes of nonnegative matrices which arise in applications (stochastic, oscillatory, totally nonnegative and \(M\)-matrices). Ch. 7 deals with inverse eigenvalue problems, inverse spectrum problems and with similarity to nonnegative and to double stochastic matrices. At the end of each chapter the reader finds problems and references.

The topic of the book is important and interesting and the book can be used by a broad audience.

Reviewer: A.G.Ramm

### MSC:

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

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

15A18 | Eigenvalues, singular values, and eigenvectors |

15B51 | Stochastic matrices |