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Curves and spectrum localization for real matrices.

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Title: Curves and spectrum localization for real matrices.
Authors: Aretaki, Aikaterini1,2 (AUTHOR) kathy@uth.gr, Adam, Maria1 (AUTHOR) madam@dib.uth.gr, Tsatsomeros, Michael3 (AUTHOR) tsat@wsu.edu
Source: Linear Algebra & its Applications. Mar2026, Vol. 733, p116-154. 39p.
Subjects: Eigenvalues, Cubic curves, Geometry, Mathematical analysis, Matrices (Mathematics), Spectrometry
Abstract: It is well known that the eigenvalues of a complex matrix A are located to the left of the vertical line passing through the largest eigenvalue of its Hermitian part, H (A). Adam and Tsatsomeros in [1] defined a cubic algebraic curve, known as the shell Γ 1 (A) of A , using the two largest eigenvalues of H (A). This curve localizes the spectrum further and lies to the left of the aforementioned vertical line. Later, Bergqvist in [5] extended the methodology employed in [1] to define a new curve, Γ 2 (A) , in terms of the three largest eigenvalues of H (A). This article delves into the geometry of Γ 2 (A) for a real matrix A to address some open questions raised in [5]. In particular, specific conditions are established to characterize the configurations of Γ 2 (A) in certain cases. Additionally, the number of eigenvalues of A surrounded by a bounded branch of the curve is examined. Examples are used to validate our findings and demonstrate the quality of Γ 2 (A) as a finer spectrum localization area when compared to Γ 1 (A). [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:It is well known that the eigenvalues of a complex matrix A are located to the left of the vertical line passing through the largest eigenvalue of its Hermitian part, H (A). Adam and Tsatsomeros in [1] defined a cubic algebraic curve, known as the shell Γ 1 (A) of A , using the two largest eigenvalues of H (A). This curve localizes the spectrum further and lies to the left of the aforementioned vertical line. Later, Bergqvist in [5] extended the methodology employed in [1] to define a new curve, Γ 2 (A) , in terms of the three largest eigenvalues of H (A). This article delves into the geometry of Γ 2 (A) for a real matrix A to address some open questions raised in [5]. In particular, specific conditions are established to characterize the configurations of Γ 2 (A) in certain cases. Additionally, the number of eigenvalues of A surrounded by a bounded branch of the curve is examined. Examples are used to validate our findings and demonstrate the quality of Γ 2 (A) as a finer spectrum localization area when compared to Γ 1 (A). [ABSTRACT FROM AUTHOR]
ISSN:00243795
DOI:10.1016/j.laa.2025.12.013