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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]
Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
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  Data: Curves and spectrum localization for real matrices.
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  Data: <searchLink fieldCode="AR" term="%22Aretaki%2C+Aikaterini%22">Aretaki, Aikaterini</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> kathy@uth.gr</i><br /><searchLink fieldCode="AR" term="%22Adam%2C+Maria%22">Adam, Maria</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> madam@dib.uth.gr</i><br /><searchLink fieldCode="AR" term="%22Tsatsomeros%2C+Michael%22">Tsatsomeros, Michael</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> tsat@wsu.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22Linear+Algebra+%26+its+Applications%22">Linear Algebra & its Applications</searchLink>. Mar2026, Vol. 733, p116-154. 39p.
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  Data: <searchLink fieldCode="DE" term="%22Eigenvalues%22">Eigenvalues</searchLink><br /><searchLink fieldCode="DE" term="%22Cubic+curves%22">Cubic curves</searchLink><br /><searchLink fieldCode="DE" term="%22Geometry%22">Geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+analysis%22">Mathematical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Spectrometry%22">Spectrometry</searchLink>
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  Data: 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]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1016/j.laa.2025.12.013
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      – Code: eng
        Text: English
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        PageCount: 39
        StartPage: 116
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      – SubjectFull: Eigenvalues
        Type: general
      – SubjectFull: Cubic curves
        Type: general
      – SubjectFull: Geometry
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      – SubjectFull: Mathematical analysis
        Type: general
      – SubjectFull: Matrices (Mathematics)
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      – SubjectFull: Spectrometry
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      – TitleFull: Curves and spectrum localization for real matrices.
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            – D: 15
              M: 03
              Text: Mar2026
              Type: published
              Y: 2026
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