arXiv subjects

Joey Veltri

Publications and source records attributed to Joey Veltri.

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

The structure of weight and function classes with coprime bases

In a recent work of Anderson and Hu, the authors constructed a measure that was $p$-adic and $q$-adic doubling, for any primes $p$ and $q$, yet not doubling. This work relied heavily on a developed number theory framework. Here we develop this framework farther, which yields a measure that is $m$-adic and $n$-adic doubling for any coprime $m,n$, yet not doubling. Additionally we show several new applications to the intersection of weight and the function classes.

math.NT↗

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↗