On universal realizability of spectra
A list $Λ=\{λ_{1},λ_{2},\ldots ,λ_{n}\}$ 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 ($\mathcal{UR}$) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by $Λ$. It is well known that an $n\times n$ 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 $ε\geq 0$ and $Λ=\{λ_{1},λ_{2},\ldots ,λ_{n}\}$ is $\mathcal{UR},$ then $\{λ_{1}+ε,λ_{2},\ldots,λ_{n}\}$ is also $\mathcal{UR}$. We give counter-examples for the cases: $Λ=\{λ_{1},λ_{2},\ldots ,λ_{n}\}$ is $\mathcal{UR}$ implies $\{λ_{1}+ε,λ_{2}-ε,λ_{3},\ldots ,λ_{n}\}$ is $\mathcal{UR},$ and $Λ_{1},Λ_{2}$ are $\mathcal{UR}$ implies $Λ_{1}\cup Λ_{2}$ is $\mathcal{UR}$.