On universal realizability of spectra

Abstract

A list $\Lambda =\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list $\Lambda$ is said to be universally realizable ($\mathcal{UR}$) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by $\Lambda$. It is well known that an $n\times n$ 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 $\epsilon \ge 0$ and $\Lambda =\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\}$ is $\mathcal{UR},$ then $\{{\lambda }_1+\epsilon ,{\lambda }_2,\dots ,{\lambda }_n\}$ is also $\mathcal{UR}$. We give counter-examples for the cases: $\Lambda =\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\}$ is $\mathcal{UR}$ implies $\{{\lambda }_1+\epsilon ,{\lambda }_2-\epsilon ,{\lambda }_3,\dots ,{\lambda }_n\}$ is $\mathcal{UR},$ and ${\Lambda }_1,{\Lambda }_2$ are $\mathcal{UR}$ implies ${\Lambda }_1\cup {\Lambda }_2$ is $\mathcal{UR}$.

22 pages, 2 figures