Open AccessOn the eigenvalues of matrices with common Gershgorin regions
Author(s) -
Anna Davis,
Paul Zachlin
Publication year - 2022
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 31
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2022.6025
Subject(s) - eigenvalues and eigenvectors , mathematics , algebraic number , gravitational singularity , variable (mathematics) , spectrum of a matrix , matrix (chemical analysis) , pure mathematics , mathematical analysis , matrix differential equation , physics , chemistry , quantum mechanics , chromatography , differential equation
This paper is a study of the eigenvalues of a complex square matrix with one variable nondiagonal entry expressed in polar form. Changing the angle of the variable entry while leaving the radius fixed generates an algebraic curve; as does the process of fixing an angle and varying the radius. The authors refer to these two curves as eigenvalue orbits and eigenvalue trajectories, respectively. Eigenvalue orbits and trajectories are orthogonal families of curves, and eigenvalue orbits are sets of eigenvalues from matrices with identical Gershgorin regions. Algebraic and geometric properties of both types of curves are examined. Features such as poles, singularities, and foci are discussed.