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November 2021 A Proof of the Elliptical Range Theorem via Kippenhahn's Theorem
Pietro Paparella, Luis J. Ramirez, Yen-Fen Wang
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Missouri J. Math. Sci. 33(2): 181-185 (November 2021). DOI: 10.35834/2021/3302181

Abstract

The elliptical range theorem asserts that the field of values (or numerical range) of a two-by-two matrix with complex entries is an elliptical disk, the foci of which are the eigenvalues of the given matrix. Many proofs of this result are available in the literature, but most, with one exception, are computational and quite involved. In this note, it is shown that the elliptical range theorem follows from the properties of plane algebraic curves and a straightforward application of a well-known result due to Kippenhahn.

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Pietro Paparella. Luis J. Ramirez. Yen-Fen Wang. "A Proof of the Elliptical Range Theorem via Kippenhahn's Theorem." Missouri J. Math. Sci. 33 (2) 181 - 185, November 2021. https://doi.org/10.35834/2021/3302181

Information

Published: November 2021
First available in Project Euclid: 30 November 2021

Digital Object Identifier: 10.35834/2021/3302181

Subjects:
Primary: 15A60

Keywords: elliptical range theorem , field of values , numerical range

Rights: Copyright © 2021 University of Central Missouri, School of Computer Science and Mathematics

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Vol.33 • No. 2 • November 2021
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