Foreword

Summary

In this monograph was present the central properties of finite totally positive matrices. As such, the monograph has only six main chapters. We consider the basic properties of such matrices, determinantal criteria for when a matrix is totally positive, their variation diminishing properties, various examples of totally positive matrices, their eigenvalue/eigenvector properties, and factorizations of such matrices. Numerous topics are excluded from this exposition. Total positivity is a theory of considerable consequence, and the most glaring omissions of this monograph are undoubtedly its various applications to diverse areas. Aside from the many applications mentioned in Gantmacher, Krein [1950] and Karlin [1968], applications can be found in approximation theory (see Schumaker [1981], Pinkus [1985c]), combinatorics (see Brenti [1989], [1995], [1996]), graph theory (see Fomin, Zelevinsky [2000], Berenstein, Fomin, Zelevinsky [1996]), Lie group theory (see Lusztig [1994]), majorization (see Marshall, Olkin [1979]), noncommutative harmonic analysis (see Gross, Richards [1995]), shape preservation (see Goodman [1995]), computing using totally positive matrices (see de Boor, Pinkus [1977], Koev [2005], Demmel, Koev [2005], Koev [2007]), refinement equations and subdivision (see Cavaretta, Dahmen, Micchelli [1991], Micchelli, Pinkus [1991]), and infinite totally positive banded matrices (see Cavaretta, Dahmen, Micchelli, Smith [1981], de Boor [1982], Smith [1983], Dahmen, Micchelli, [1986]). See also the many references in these papers and also the many references to these papers. There has been no attempt to make this monograph all-encompassing, and we apologize to all who feel that their contributions to the theory have been slighted as a consequence.