2013/06/13 by Richard Ellard, Ellard, Richard, Helena Šmigoc +1
Computer Science · Mathematics · #15A18 #15A29 #Advanced Algebra and Logic #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1306.2998
openalex publication_date 2013/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a list of complex numbers σ:=(λ1,λ2,...,λm), we say that σ is realisable if σ 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 (ρ,λ2,λ3,...,λm), where ρ is the Perron eigenvalue and λ2 is real, we find families of lists (μ1,μ2,...,μn), for which (μ1,μ2,...,μn,λ3,λ4,...,λm) is realisable. In addition, given a realisable list (ρ,α+iβ,α-iβ,λ4,λ5,...,λm), where ρ is the Perron eigenvalue and α and β are real, we find families of lists (μ1,μ2,μ3,μ4), for which (μ1,μ2,μ3,μ4,λ4,λ5,...,λm) is realisable.