The Electronic Journal of Linear Algebra

A permanent inequality for positive semidefinite matrices

Vehbi E. Paksoy — 2023-02-24

In this paper, we prove an inequality involving the permanent of a positive semidefinite matrix and its leading submatrices. We obtain a result in the similar spirit of Bapat-Sunder per-max conjecture.

Linear maps that preserve parts of the spectrum on pairs of similar matrices

Constantin Costara — 2023-03-23

In this paper, we characterize linear bijective maps $\varphi$ on the space of all $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ having the property that the spectrum of $\varphi (A)$ and $\varphi (B)$ have at least one common eigenvalue for each similar matrices $A$ and $B$. Using this result, we characterize linear bijective maps having the property that the spectrum of $\varphi (A)$ and $\varphi (B)$ have common elements for each matrices $A$ and $B$ having the same spectrum. As a corollary, we also characterize linear bijective maps preserving the equality of the spectrum.

The inverse Horn problem

João Filipe Queiró — 2023-03-01

Alfred Horn's conjecture on eigenvalues of sums of Hermitian matrices was proved more than 20 years ago. In this note, the problem is raised of, given an $n$-tuple $\gamma$ in the solution polytope, constructing Hermitian matrices with the required spectra such that their sum has eigenvalues $\gamma$.

Hypocoercivity and hypocontractivity concepts for linear dynamical systems

Franz Achleitner — 2023-02-10

For linear dynamical systems (in continuous-time and discrete-time), we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time behavior of the propagator norm for linear continuous-time and discrete-time systems is characterized by the (shifted) hypocoercivity index and the (scaled) hypocontractivity index, respectively.

An improved algorithm for solving an inverse eigenvalue problem for band matrices

Kanae Akaiwa — 2022-12-03

The construction of matrices with prescribed eigenvalues is a kind of inverse eigenvalue problems. The authors proposed an algorithm for constructing band oscillatory matrices with prescribed eigenvalues based on the extended discrete hungry Toda equation (Numer. Algor. 75:1079--1101, 2017). In this paper, we develop a new algorithm for constructing band matrices with prescribed eigenvalues based on a generalization of the extended discrete hungry Toda equation. The new algorithm improves the previous algorithm so that the new one can produce more generic band matrices than the previous one in a certain sense. We compare the new algorithm with the previous one by numerical examples. Especially, we show an example of band oscillatory matrices which the new algorithm can produce but the previous one cannot.