Constructing New Realisable Lists from Old in the NIEP

TL;DR

This paper develops methods to construct new realisable lists of eigenvalues from existing ones within the Nonnegative Inverse Eigenvalue Problem, expanding the known classes of spectra that can be realized by nonnegative matrices.

Contribution

It introduces new techniques to generate larger realisable lists from known spectra, specifically handling cases with Perron eigenvalues and complex conjugates.

Findings

Established families of realisable lists with specified eigenvalue structures

Extended the set of known spectra that can be realized by nonnegative matrices

Provided constructive methods for spectrum augmentation

Abstract

Given a list of complex numbers \sigma:=(\lambda_1,\lambda_2,...,\lambda_m), we say that {\sigma} is realisable if {\sigma} is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (or NIEP) is the problem of categorising all realisable lists. Given a realisable list (\rho,\lambda_2,\lambda_3,...,\lambda_m), where {\rho} is the Perron eigenvalue and \lambda_2 is real, we find families of lists (\mu_1,\mu_2,...,\mu_n), for which (\mu_1,\mu_2,...,\mu_n,\lambda_3,\lambda_4,...,\lambda_m) is realisable. In addition, given a realisable list (\rho,\alpha+i\beta,\alpha-i\beta,\lambda_4,\lambda_5,...,\lambda_m), where {\rho} is the Perron eigenvalue and {\alpha} and {\beta} are real, we find families of lists (\mu_1,\mu_2,\mu_3,\mu_4), for which (\mu_1,\mu_2,\mu_3,\mu_4,\lambda_4,\lambda_5,...,\lambda_m) is realisable.