Nonnegative matrices with prescribed spectra

Nonnegative matrices are rectangular arrays of nonnegative real numbers. Nonnegative matrices are prevalent in many areas of study. Statistics, Economics, and Chemistry are examples of disciplines that use nonnegative matrices. In this dissertation we examined and produced new results on nonnegative matrices with prescribed spectra. This problem is called the nonnegative inverse eigenvalue problem (NIEP). The NIEP asks which lists of n complex numbers occur as the spectrum of an n x n, entry-wise nonnegative matrix. A recent paper by Leal-Duarte and Johnson stated, “this problem has attracted considerable attention over 50 years and, despite many exciting partial results, remains quite unresolved.” In this dissertation we solved the NIEP for certain lists of four complex numbers.; Since the general NIEP is difficult to resolve, two natural variations were considered. The real nonnegative inverse eigenvalue problem (RNIEP) asks which lists of n real numbers occur as the spectrum of a nonnegative n x n matrix. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which lists of n real numbers occur as the spectrum of a nonnegative n x n matrix. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which lists of n real numbers occur as the spectrum of a symmetric nonnegative n x n matrix. We resolved the SNIEP in most of the cases for lists of five real numbers and lists of six real numbers. In a paper by Johnson, Laffey, and Loewy, it was shown that the RNIEP and the SNIEP are different problems. We have shown that the smallest list size in which the RNIEP and the SNIEP are different is five.; A matrix A = [a_ij] is said to be subordinate to a given graph G if whenever vertex v_i and v_j are not adjacent then entry a_ij = 0. We resolved some cases of the NIEP where the realizing nonnegative matrix was subordinate to a bipartite graph.; In conclusion, the results in this dissertation have taken us several steps closer to the resolution of one of the outstanding problems in matrix theory.