Nonnegative matrices and applicable topics in linear algebra.

*(English)*Zbl 0625.15011
Mathematics and its applications. Chichester: Ellis Horwood Limited; New York etc.: Halsted Press: a division of John Wiley & Sons. 264 p.; £35.00 (1987).

The title of this book suggests that it needs to be viewed in the light of earlier books substantially or totally devoted to the topic of non- negative matrices (matrices with non-negative elements), in particular, the rewiever [Non-Negative Matrices and Markov Chains (1981; Zbl 0471.60001), 1st edn; Non-Negative Matrices (1973; Zbl 0278.15011)]. A. Berman and R. J. Plemmons [Non-Negative Matrices in the Mathematical Sciences (1979; Zbl 0484.15016)] as well as earlier books by Gantmacher, Marcus and Minc, and Varga. Unlike all of these, this is an undergraduate text which its author intends to “be understood by readers who are not necessarily expert mathematicians”. In line with this intention, and in constrast to earlier books, the important concepts in the text are illustrated by simple worked examples, and problems are given at the end of all but two chapters, with solutions at the end of the book.

The author feels that since “there is little point in investigating the characteristics of one family of matrices, however simple and beautiful they may be, without an investigation of some of the other families of matrices, Chapters 2 and 3 are written to do just that”. Thus after an Introductory Chapter 1, on topics such as the derivative of a determinant, the adjoint of a characteristic matrix, permutation matrices, and some aspects of the theory of graphs, Chapter 2 (“Some Matrix Types”) covers various aspects of the theory of normal matrices, including unitary and Hermitian matrices, while Chapter 3 is devoted to some of the properties of positive-definite matrices. While there are some substantial similarities between symmetric positive-definite matrices and square non-negative matrices, this reviewer remains unconvinced that Chapters 2 and 3 are more than peripherally necessary in even an elementary treatment of non-negative matrices. Chapter 4 covers the main topic of the book, non-negative matrices, and is in spirit quite close to Chapter 1 of the reviewer’s (1973) (1981) books. Chapter 5 deals with M-matrices and Chapter 6 with Markov chains and stochastic matrices. Both are, in the spirit of ealier books, an application of the results of Chapter 4. Chapter 7 contains further applications of non-negative matrices.

There are 23 books and 27 papars in the reference list at the end of the book, but referencing in the text is minimal: Chapter 4, for example, begins with a reference only to the classical paper of H. Wielandt [Math. Z. 52, 642-648 (1950; Zbl 0035.291)]. There is a subject index. The book fulfils its aims.

The author feels that since “there is little point in investigating the characteristics of one family of matrices, however simple and beautiful they may be, without an investigation of some of the other families of matrices, Chapters 2 and 3 are written to do just that”. Thus after an Introductory Chapter 1, on topics such as the derivative of a determinant, the adjoint of a characteristic matrix, permutation matrices, and some aspects of the theory of graphs, Chapter 2 (“Some Matrix Types”) covers various aspects of the theory of normal matrices, including unitary and Hermitian matrices, while Chapter 3 is devoted to some of the properties of positive-definite matrices. While there are some substantial similarities between symmetric positive-definite matrices and square non-negative matrices, this reviewer remains unconvinced that Chapters 2 and 3 are more than peripherally necessary in even an elementary treatment of non-negative matrices. Chapter 4 covers the main topic of the book, non-negative matrices, and is in spirit quite close to Chapter 1 of the reviewer’s (1973) (1981) books. Chapter 5 deals with M-matrices and Chapter 6 with Markov chains and stochastic matrices. Both are, in the spirit of ealier books, an application of the results of Chapter 4. Chapter 7 contains further applications of non-negative matrices.

There are 23 books and 27 papars in the reference list at the end of the book, but referencing in the text is minimal: Chapter 4, for example, begins with a reference only to the classical paper of H. Wielandt [Math. Z. 52, 642-648 (1950; Zbl 0035.291)]. There is a subject index. The book fulfils its aims.

Reviewer: Eugene Seneta (Sydney)

##### MSC:

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

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

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

15B51 | Stochastic matrices |