Spectral properties of nonnegative matrices

The spectral properties of nonnegative matrices have intrigued pure and applied mathematicians alike, beginning with the classical works of Oskar Perron and Georg Frobenius at the start of the twentieth century. One question which stems naturally from this area of research is that of the "Nonnegative Inverse Eigenvalue Problem", or NIEP. This is the problem of characterising those lists of complex numbers which are "realisable" as the spectrum of some entrywise nonnegative matrix. This thesis explores the NIEP, as well as one of its variants, the "Symmetric Nonnegative Inverse Eigenvalue Problem", or SNIEP, which considers realisability by a symmetric nonnegative matrix.The question of determining which operations on lists preserve realisability is pertinent in the NIEP, since such operations can allow us to construct more complicated lists from simple building blocks. We present some new results along these lines. In particular, we discuss how to replace parts of realisable lists by longer lists, while preserving realisability.In those cases where a realising matrix is known to exist, one can consider studying the properties of this matrix. We focus our attention on the problem of characterising the diagonal elements of the realising matrix and achieve a complete solution in the case where every entry in the list (apart from the Perron eigenvalue) has nonpositive real part. In order to prove this result, we derived complex analogues of Newton's inequalities, which are of independent interest.In the context of the SNIEP, we unify a large body of research by presenting a recursive method for constructing symmetrically realisable lists and showing that essentially all previously know sufficient conditions are either contained in, or equivalent to the family we introduce. Our construction also reveals several interesting properties of the family in question and allows for an explicit algorithmic characterisation of the lists that lie within it.Finally, we construct families of symmetrically realisable lists which do not satisfy any previously known sufficient conditions.