arXiv subjects

Ricardo L. Soto

Publications and source records attributed to Ricardo L. Soto.

8 recordsLinked to original sources

arXiv2023

Indices of diagonalizable and universal realizability of spectra

A list $Λ=\{λ_{1},\ldots ,λ_{n}\}$ of complex numbers (repeats allowed) is said to be \textit{realizable} if it is the spectrum of an entrywise nonnegative matrix $A$. $Λ$ is \textit{diagonalizably realizable} if the realizing matrix $A$ is diagonalizable. $Λ$ is said to be \textit{universally realizable} if it is \textit{\ realizable} for each possible Jordan canonical form allowed by $Λ.$ Here, we study the connection between diagonalizable realizability and universal realizability of spectra. In particular, we establish \textit{\ indices of realizability} for diagonalizable and universal realizability. We also define the merge of two spectra and we prove a result that allow us to easily decide, in many cases, about the universal realizability of spectra.

math.SP↗

arXiv2023

Smigoc's glue for universal realizability in the left half-plane

A list of A list Λ of complex numbers is said to be realizable if it is the spectrum of a nonnegative matrix. Λ is said to be universally realizable (UR) if it is realizable for each possible Jordan canonical form allowed by Λ. In this paper, using companion matrices and applying a procedure by Šmigoc, is provides a sufficient condition for the universal realizability of left half-plane spectra. It is also shown how the effect of adding a negative real number to a not UR left half-plane list of complex numbers, makes the new list UR, and a family of left half-plane lists that are UR is characterized.

math.SP↗

arXiv2020

The role of certain Brauer and Rado results in the nonnegative inverse spectral problems

We say that a list $Λ=\{ λ_{1},\ldots ,λ_{n}\}$ of complex numbers is realizable, if it is the spectrum of a nonnegative matrix $A$ (the realizing matrix). We say that $Λ$ is universally realizable if it is realizable for each possible Jordan canonical form allowed by $ Λ.$ This work does not contain new results. As its title says, our goal is to show and emphasize the relevance of certain results of Brauer and Rado in the study of nonnegative inverse spectral problems. We show that virtually all known results, which give sufficient conditions for the list $ Λ$ to be realizable or universally realizable, can be obtained from the results of Brauer or Rado. Moreover, in this case, we may always compute a realizing matrix.

math.SP↗

arXiv2019

Centrosymmetric nonnegative realization 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. In this paper we intent to characterize those lists of complex numbers, which are realizable by a centrosymmetric nonnegative matrix. In particular, we show that lists of nonnegative real numbers, and lists of complex numbers of Suleimanova type (except in one particular case), are always the spectrum of some centrosymmetric nonnegative matrix. For the general lists we give sufficient conditions via a perturbation result. We also show that for $n=4,$ every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix.

math.RA↗

arXiv2018

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 $\mathcal{UR}$.

math.SP↗

arXiv2018

On Fillmore's theorem extended by Borobia

Fillmore Theorem says that if A is an nxn complex non-scalar matrix and γ_1,...,γ_{n} are complex numbers with γ_1+...+γ_{n}=trA, then there exists a matrix B similar to A with diagonal entries γ_1,...,γ_{n}. Borobia simplifies this result and extends it to matrices with integer entries. Fillmore and Borobia do not consider the nonnegativity hypothesis. Here, we introduce a different and very simple way to compute the matrix B similar to A with diagonal γ_1,...,γ_{n}. Moreover, we consider the nonnegativity hypothesis and we show that for a list Λ={λ_1,...,λ_{n}} of complex numbers of Suleimanova or Šmigoc type, and a given list Γ={γ_1,...,γ_{n}} of nonnegative real numbers, the remarkably simple condition γ_1+...+γ_{n}=λ_1+...+λ_{n} is necessary and sufficient for the existence of a nonnegative matrix with spectrum Λ and diagonal entries Γ. This surprising simple result improves a condition recently given by Ellard and Šmigoc in arXiv:.1702.02650v1.

math.SP↗

arXiv2017

Permutative nonnegative matrices with prescribed spectrum

An n x n permutative matrix is a matrix in which every row is a permutation of the first row. In this paper the result given by Paparella in [Electron. J. Linear Algebra 31 (2016) 306-312] is extended to a more general lists of real and complex numbers, and a negative answer to a question posed by him is given.

math.SP↗

arXiv2016

On the $A_α$-spectra of trees

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $α\in\left[ 0,1\right],$ define the matrix $A_α\left(G\right) $ as \[ A_α\left(G\right) =αD\left(G\right) +(1-α)A\left(G\right) \] where $0\leqα\leq1$. This paper gives several results about the $A_α$-matrices of trees. In particular, it is shown that if $T_Δ$ is a tree of maximal degree $Δ,$ then the spectral radius of $A_α(T_Δ)$ satisfies the tight inequality \[ ρ(A_α(T_Δ))<αΔ+2(1-α)\sqrt{Δ-1}. \] This bound extends previous bounds of Godsil, Lovász, and Stevanović. The proof is based on some new results about the $A_α$-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of $A_α$ of general graphs are proved, implying tight bounds for paths and Bethe trees.

math.CO↗