Totally positive matrices constitute a particular class of matrices, the study of which was initiated by analysts because of its many applications in diverse areas. This account of the subject is comprehensive and thorough, with careful treatment of the central properties of totally positive matrices, full proofs and a complete bibliography. The history of the subject is also described: in particular, the book ends with a tribute to the four people who have made the most notable contributions to the history of total positivity: I. J. Schoenberg, M. G. Krein, F. R. Gantmacher and S. Karlin. This monograph will appeal to those with an interest in matrix theory, to those who use or have used total positivity, and to anyone who wishes to learn about this rich and interesting subject.
Foreword; 1. Basic properties of totally positive matrices; 2. Criteria for total positivity and strict total positivity; 3. Variation diminishing; 4. Examples; 5. Eigenvalues and eigenvectors; 6. Factorizations of totally positive matrices; Afterword; References; Subject index; Author index.
"... found the book particularly effective as a tool for a reading group on algebraic statistics; it is an excellent resource, cuts straight to the subject's open questions, and should be an interesting read for any researcher for theoretical statistics."
Robin J. Evans, SIAM Review