arXiv subjects

M. I. Bueno

Publications and source records attributed to M. I. Bueno.

3 recordsLinked to original sources

arXiv2022

Linear maps preserving the Lorentz spectrum of $3 \times 3$ matrices

For a given $3 \times 3$ real matrix $A$, the eigenvalue complementarity problem relative to the Lorentz cone consists of finding a real number $λ$ and a nonzero vector $x \in \mathbb{R}^3$ such that $x^T(A-λI)x=0$ and both $x$ and $(A-λI)x$ lie in the Lorentz cone, which is comprised of all vectors in $\mathbb{R}^3$ forming a $45^\circ$ or smaller angle with the positive $z$-axis. We refer to the set of all solutions $λ$ to this eigenvalue complementarity problem as the Lorentz spectrum of $A$. Our work concerns the characterization of the linear preservers of the Lorentz spectrum on the space $M_3$ of $3 \times 3$ real matrices, that is, the linear maps $ϕ: M_3 \to M_3$ such that the Lorentz spectra of $A$ and $ϕ(A)$ are the same for all $A$. We have proven that all such linear preservers take the form $ϕ(A) = (Q \oplus [1])A(Q^T \oplus [1])$, where $Q$ is an orthogonal $2 \times 2$ matrix.

math.RA↗

arXiv2022

Linear maps preserving the Lorentz spectrum: The $2\times 2$ case

In this paper a complete description of the linear maps $ϕ:W_{n}\rightarrow W_{n}$ that preserve the Lorentz spectrum is given when $n=2$ and $W_{n}$ is the space $M_{n}$ of $n\times n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it has been shown that $ϕ(A)=PAP^{-1}$ for all $A\in W_{2}$, where $P$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lorentz cone). These results extend to $n=2$ those for $n\geq 3$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $n\geq3,$ due to the fact that the Lorentz cone in $\mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in $\mathbb{R}^{n}$ with $n\geq3.$ Thus, the study of the Lorentz spectrum preservers on $W_n = M_n$ also follows from the known description of the Pareto spectrum preservers on $M_n$.

math.RA↗

arXiv2017

Conditioning and backward error of block-symmetric block-tridiagonal linearizations of matrix polynomials

For each square matrix polynomial $P(λ)$ of odd degree, a block-symmetric block-tridiagonal pencil $\mathcal{T}_{P}(λ)$ was introduced by Antoniou and Vologiannidis in 2004, and a variation $\mathcal{R}_P(λ)$ was introduced by Mackey et al. in 2010. These two pencils have several appealing properties, namely they are always strong linearizations of $P(λ)$, they are easy to construct from the coefficients of $P(λ)$, the eigenvectors of $P(λ)$ can be recovered easily from those of $\mathcal{T}_P(λ)$ and $\mathcal{R}_P(λ)$, the two pencils are symmetric (resp. Hermitian) when $P(λ)$ is, and they preserve the sign characteristic of $P(λ)$ when $P(λ)$ is Hermitian. In this paper we study the numerical behavior of $\mathcal{T}_{P}(λ)$ and $\mathcal{R}_P(λ)$. We compare the conditioning of a finite, nonzero, simple eigenvalue $δ$ of $P(λ)$, when considered an eigenvalue of $P(λ)$ and an eigenvalue of $\mathcal{T}_{P}(λ)$. We also compare the backward error of an approximate eigenpair $(z,δ)$ of $\mathcal{T}_{P}(λ)$ with the backward error of an approximate eigenpair $(x,δ)$ of $P(λ)$, where $x$ was recovered from $z$ in an appropriate way. When the matrix coefficients of $P(λ)$ have similar norms and $P(λ)$ is scaled so that the largest norm of the matrix coefficients of $P(λ)$ is one, we conclude that $\mathcal{T}_{P}(λ)$ and $\mathcal{R}_P(λ)$ have good numerical properties in terms of eigenvalue conditioning and backward error. Moreover, we compare the numerical behavior of $\mathcal{T}_{P}(λ)$ with that of other well-studied linearizations in the literature, and conclude that $\mathcal{T}_{P}(λ)$ performs better than these linearizations when $P(λ)$ has odd degree and has been scaled.

math.NA↗