Complex Eigenvalue Analysis of a Linear Pencil: An Experimental Study

Year 2024, Volume: 16 Issue: 2, 518 - 528, 31.12.2024

Hasen Mekki Öztürk

https://doi.org/10.47000/tjmcs.1349180

Abstract

This paper is concerned with a finite-dimensional example of a linear pencil which leads to a class of non-self-adjoint matrices. We consider the linear pencil $H_c-\lambda L$, where $H_c$ is a tri-diagonal matrix with a constant parameter $c$ on the main diagonal and off-diagonal entries equal to one, and $L$ is a diagonal matrix whose elements decrease linearly from one to minus one. In general, the spectra of operator polynomials may contain non-real eigenvalues as well as real eigenvalues. Nevertheless, they exhibit certain patterns. Our aim in this research is to carry out a variety of numerical investigation on the eigenvalues so as to understand the eigenvalue behaviour of such pencils from different points of view. In accordance with our numerical findings, a series of conjectures are offered and various heuristics has been discussed.

Keywords

Linear operator pencils, non-self-adjoint matrices, eigenvalues, spectral theory

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