2018/09/06 by Ana I. Julio, Julio, Ana I., Carlos Marijuán +5
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1809.02224
openalex publication_date 2018/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A list Λ=\λ1,λ2,… ,λn\ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list Λ is said to be universally realizable (UR) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an n× n nonnegative matrix A is co-spectral to a nonnegative matrix B with constant row sums. In this paper, we extend the co-spectrality between A and B to a similarity between A and B, when the Perron eigenvalue is simple. We also show that if ε≥ 0 and Λ=\λ1,λ2,… ,λn\ is UR, then \λ1+ε,λ2,…,λn\ is also UR. We give counter-examples for the cases: Λ=\λ1,λ2,… ,λn\ is UR implies \λ1+ε,λ2-ε,λ3,… ,λn\ is UR, and Λ1,Λ2 are UR implies Λ1∪ Λ2 is UR.