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On numerical ranges of the compressions of normal matrices

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Title: On numerical ranges of the compressions of normal matrices
Authors: Adam, Maria1 madam@ucg.gr
Source: Applied Mathematics & Computation. Jan2011, Vol. 217 Issue 9, p4699-4709. 11p.
Subjects: Numerical analysis, Matrices (Mathematics), Isometrics (Mathematics), Projective geometry, Boundary value problems, Mathematical analysis
Abstract: Abstract: For an n × n normal matrix A, whose numerical range NR[A] is a k-polygon (k ⩽ n), an n ×(k −1) isometry matrix P is constructed by a unit vector , and NR[P∗AP] is inscribed to NR[A]. In this paper, using the notations of NR[P∗AP] and some properties from projective geometry, an n × n diagonal matrix B and an n ×(k −2) isometry matrix Q are proposed such that NR[P∗AP] and NR[Q∗BQ] have as common support lines the edges of the k-polygon and share the same boundary points with the polygon. It is proved that the boundary of NR[P∗AP] is a differentiable curve and the boundary of the numerical range of a 3×3 matrix P∗AP is an ellipse, when the polygon is a quadrilateral. [ABSTRACT FROM AUTHOR]
Copyright of Applied Mathematics & Computation 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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  Data: Abstract: For an n × n normal matrix A, whose numerical range NR[A] is a k-polygon (k ⩽ n), an n ×(k −1) isometry matrix P is constructed by a unit vector , and NR[P∗AP] is inscribed to NR[A]. In this paper, using the notations of NR[P∗AP] and some properties from projective geometry, an n × n diagonal matrix B and an n ×(k −2) isometry matrix Q are proposed such that NR[P∗AP] and NR[Q∗BQ] have as common support lines the edges of the k-polygon and share the same boundary points with the polygon. It is proved that the boundary of NR[P∗AP] is a differentiable curve and the boundary of the numerical range of a 3×3 matrix P∗AP is an ellipse, when the polygon is a quadrilateral. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Applied Mathematics & Computation 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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        Value: 10.1016/j.amc.2010.11.023
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        Text: English
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      – SubjectFull: Isometrics (Mathematics)
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      – SubjectFull: Projective geometry
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      – SubjectFull: Boundary value problems
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      – SubjectFull: Mathematical analysis
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