Ph.D Student Ph.D Thesis Spector Oren The Symmetric Nonnegative Inverse Eigenvalue Problem Department of Mathematics PROFESSOR EMERITUS Raphael Loewy

Abstract

Nonnegative matrices appear naturally in different fields of the mathematical sciences, such as probability, engineering, and economics. It turns out that nonnegative square matrices have special spectral properties. The fundamental theorem of the spectral theory of nonnegative matrices is the Perron-Frobenius theorem, which is about 100 years old.

One of the most difficult problems in matrix theory is to determine the lists of n complex numbers (respectively real numbers) which are the spectra of nn nonnegative (respectively symmetric nonnegative) matrices, called the Nonnegative (respectively Symmetric Nonnegative) Inverse Eigenvalue Problem. In fact, this problem is open for any n≥5, despite the fact that more than 60 years have passed since it was formulated by Suleimanova. The problems are fully solved only in the special case when n≤4 and are partially solved in the case n=5.

Our work deals with the first open case, that is n=5, for a list of real numbers. We made a significant progress towards the solution of this case. In particular, we obtain the solution when the sum of the five given numbers is zero or at least half of the largest one. In the latter case we also find a necessary condition the eigenvalues of symmetric nonnegative 5⨯5 matrices must satisfy.