Totally Nonnegative Matrices
Shaun Fallat
College of William & Mary, Department of Applied Science, 1999
Field: Applied Mathematics, Degree: Ph.D.
Advisor: Charles R. Johnson, Class of 1961 Professor of Mathematics
Abstract
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.
