Matrix Positivity

Description

Matrix positivity is a central topic in matrix theory: properties that generalize the notion of positivity to matrices arose from a large variety of applications, and many have also taken on notable theoretical significance, either because they are natural or unifying. This is the first book to provide a comprehensive and up-to-date reference of important material on matrix positivity classes, their properties, and their relations. The matrix classes emphasized in this book include the classes of semipositive matrices, P-matrices, inverse M-matrices, and copositive matrices. This self-contained reference will be useful to a large variety of mathematicians, engineers, and social scientists, as well as graduate students. The generalizations of positivity and the connections observed provide a unique perspective, along with theoretical insight into applications and future challenges. Direct applications can be found in data analysis, differential equations, mathematical programming, computational complexity, models of the economy, population biology, dynamical systems and control theory.

Charles R. Johnson is Class of 1961 Professor of Mathematics at College of William and Mary. He received his B.A. in mathematics and economics from Northwestern University in 1969 and his Ph.D. from The California Institute of Technology in 1972. He received tenure at the University of Maryland, College Park in 1976, had a brief professorship at Clemson University, and has been at College of William and Mary since 1987. He has had 13 PhD students, 6 Master's students, and involved more than 200 undergraduates in his research (steadily supported by NSF). Working in most parts of matrix analysis, and especially its interface with combinatorics, Professor Johnson has published nearly 500 papers and 15 books, and received several prizes. Ronald L. Smith is Professor Emeritus at the University of Tennessee, Chattanooga. He received his PhD in Mathematics at Auburn University, as well as an M.S. in Industrial Engineering at Auburn University and an M.S. in Mathematics at the University of California, Riverside. He taught at Northern Arizona State University for 4 years, and was Professor at University of Tennessee, Chattanooga for 35 years. Dr. Smith's research interests are linear algebra, graph theory, and matrix theory, with particular interest in nonnegative matrices and matrix completions. He received the Thomson-Reuters Original List of Highly Cited Researchers Award. The conference, "Recent Advances in Linear Algebra and Graph Theory" was held in his honour. Michael J. Tsatsomeros is Professor of Mathematics at Washington State University. He received his PhD in Mathematics at the University of Connecticut in 1990. He has held positions at the University of Victoria, University of Wisconsin-Madison, University of Regina, and the College of William and Mary, and has served as Professor at Washington State University since 2007. Professor Tsatsomeros has also served the International Linear Algebra Society in various capacities, including as a member of the Board of Directors, Advisory Committee, and Journal Committee. He is Co-Editor-In-Chief of the Electronic Journal of Linear Algebra and Associate Editor of Linear Algebra and Its Applications. His main area of research is nonnegative matrix theory and generalizations.

'When one thinks of positive matrices, usually only entrywise positive matrices and positive definite matrices come to mind. This book compiles results about an amazing array of 'positive' matrices beyond such familiar ones; semipositive matrices, inverse M-matrices and copositive matrices, to name a few. The treatment is lucid and a pleasure to read. Will facilitate anyone to widen the knowledge of matrix classes.' R. B. Bapat, Indian Statistical Institute 'Positivity is one of the central topics in mathematics. Positivity of matrices is a rich and interesting research area of linear algebra and combinatorial matrix theory. Exhibiting many positivity classes of matrices with diligence, this monograph will be a very useful reference in research and applications.' Fuzhen Zhang, Nova Southeastern University '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.' Brian Borchers, MAA Reviews