Patterned matrices (T. Laffey)

Suppose σ = (λ₁, ..., λ_n) is a list of complex numbers. We say that σ is realizable if σ is the spectrum of an (entrywise) nonnegative matrix. In this case, we say that A realizables σ. Suppose that σ satisfies all known necessary conditions for realizability. Then one seeks to prove that σ is realizable by constructing a realizing matrix. Several authors have sought realizing matrices A whose entries exhibit some nice patterns, e.g. companion matrices, circulants, upper Hessenberg matrices, M-matrices. Frequently, one seeks patterned matrices A whose characteristic polynomial is easy to express in terms of the entries of A. Most known examples are of block-companion type, and we seek other useful patterns. In particular, identifying subclasses of the class of symmetric matrices which lead to realizations of real spectra has proved very elusive. We aim to identify further useful classes.