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Matrix Positivity

Charles R. Johnson, Ronald L. Smith, and Michael J. Tsatsomeros
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Tracts in Mathematics
[Reviewed by
Brian Borchers
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This book is a collection of definitions and results on various notions of positivity of matrices.  The classes of matrices discussed in the book include the familiar classes of nonnegative matrices and positive definite matrices, as well as less well-known classes of matrices such as P-matrices, M-matrices, copositive matrices, and completely positive matrices.  These classes of matrices have applications in many areas of mathematics and statistics.
In the first two chapters of the book, the authors define the various kinds of matrix positivity and give relationships between the classes. The remainder of the book is a collection of theorems on the various classes of matrices.  Proofs of some of these theorems are included, but in many cases, the authors simply cite papers where the results have been proven.  In addition to the theorems, the authors also include a number of conjectures and unsolved problems.  Discussion of applications is fairly limited.  A very thorough bibliography includes 18 pages of references.  However, the index is somewhat sparse at 3 pages.
Matrix Positivity is a reference work that will be useful not only to researchers and graduate students working in the area but also to readers who wish to find and apply results on matrix positivity to other areas of research.  For readers who want a textbook introduction to matrix analysis, Charles R. Johnson has coauthored a pair of excellent textbooks with Roger Horn, Matrix Analysis, and Topics in Matrix Analysis.
Brian Borchers is a professor of Mathematics at New Mexico Tech and the editor of MAA Reviews.