Totally nonnegative matrices

An m-by-n matrix A is called totally nonnegative (resp. totally positive) if the determinant of every square submatrix (i.e., minor) of A is nonnegative (resp. positive). The class of totally nonnegative matrices has been studied considerably, and this class arises in a variety of applications such as differential equations, statistics, mathematical biology, approximation theory, integral equations and combinatorics. The main purpose of this thesis is to investigate several aspects of totally nonnegative matrices such as spectral problems, determinantal inequalities, factorizations and entry-wise products. It is well-known that the eigenvalues of a totally nonnegative matrix are nonnegative. However, there are many open problems about what other properties exist for the eigenvalues of such matrices. In this thesis we extend classical results concerning the eigenvalues of a totally nonnegative matrix and prove that the positive eigenvalues of an irreducible totally nonnegative matrix are distinct. We also demonstrate various new relationships between the sizes and the number of Jordan blocks corresponding to the zero eigenvalue of an irreducible totally nonnegative matrix. These relationships are a necessary first step to characterizing all possible Jordan canonical forms of totally nonnegative matrices. Another notion investigated is determinantal inequalities among principal minors of totally nonnegative matrices. A characterization of all inequalities that hold among products of principal minors of totally nonnegative matrices up to at most 5 indices is proved, along with general conditions which guarantee when the product of two principal minors is less than another product of two principal minors. A third component of this thesis is a study of entry-wise products of totally nonnegative matrices. In particular, we consider such topics as: closure under this product, questions related to zero/non-zero patterns, and determinantal inequalities associated with this special product. Finally, a survey of classical results and recent developments, including: commonalities and differences among totally nonnegative matrices and other positivity classes of matrices; perturbations and factorizations of totally nonnegative matrices, are discussed