M.Sc Thesis
M.Sc Student Nathan Keller Positivity of Principal Minors and Sign Symmetry Department of Mathematics Professor Emeritus Hershkowitz Daniel

Abstract

The thesis deals with positivity of principal minors, sign symmetry (meaning that the product of minors which are symmetric with respect to the main diagonal is nonnegative), positive stability (meaning that all the eigenvalues are in the open right half-plane) and the relation between these properties.

These properties are shared by some known classes of matrices, such as totally positive matrices and the positive definite matrices, and our research is a part of a wider research which deals with determining the common properties of those classes.

P-matrices (that is, matrices all of whose principal minors are positive) were first introduced by Fiedler and Ptak in 1962. They are used in economics, physics etc.

Positive stability, as well as other types of stability, plays an important role in dynamical systems and other problems and was investigated extensively over the last century.

Sign symmetry, which is a generalization of symmetry, is relatively less known.

Carlson was the first to investigate the relation between these properties. He proved (in 1974) that a sign-symmetric P-matrix is necessarily positive stable and conjectured that even a weaker assumption, that the matrix is a weakly sign-symmetric P-matrix assures stability. His conjecture was disproved by Holtz in 1999.

In this thesis we discuss possible generalizations of Carlson’s theorem.

We formulate and prove an inverse theorem to Carlson’s theorem: a sign symmetric stable matrix is necessarily a P-matrix. We use our theorem to prove results which are related to various types of stability.

In addition, we study the relation between P-matrices and Q-matrices (that is, matrices whose sums of minors of the same order are positive). This research is based on a theorem of Kellogg (in 1972) claiming that a set of numbers is the spectrum of some P-matrix if and only if it is spectra of a Q-matrix. We find classes of matrices in which being a P-matrix is equivalent to being a Q-matrix and use this property to study the structure of those sets. We study the possibility of strengthening Carlson’s theorem by exchanging the P-matrix assumption by an assumption that the matrix is a Q-matrix. We prove the new version in partial cases and disprove it in most of the cases.

We also deal with P-matrices whose powers are also P-matrices, following works of Hershkowitz and Johnson. We study the possibility of exchanging the sign-symmetry assumption in Carlson’s theorem by an assumption that the square of the matrix is also a P-matrix and prove the new version of the theorem in partial cases.