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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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New proofs and extensions of Sylvester’s and Johnson’s inertia theorems to non-Hermitian matrices
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by Man Kam Kwong and Anton Zettl PDF
Proc. Amer. Math. Soc. 139 (2011), 3795-3806 Request permission

Abstract:

We present a new proof and extension of the classical Sylvester Inertia Theorem to a pair of non-Hermitian matrices which satisfies the property that any real linear combination of the pair has only real eigenvalues. In the proof, we embed the given problem in a one-parameter family of related problems and examine the eigencurves of the family. The proof requires only elementary matrix theory and the Intermediate Value Theorem. The same technique is then used to extend Johnson’s extension of Sylvester’s Theorem on possible values of the inertia of a product of two matrices.
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Additional Information
  • Man Kam Kwong
  • Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong
  • MR Author ID: 108745
  • ORCID: 0000-0003-0808-0925
  • Email: mankwong@polyu.edu.hk
  • Anton Zettl
  • Affiliation: Department of Mathematics, Northern Illinois University, DeKalb, Illinois 60115
  • Email: zettl@math.niu.edu
  • Received by editor(s): August 21, 2010
  • Published electronically: June 28, 2011
  • Additional Notes: Research of the first author is supported by the Hong Kong Research Grant Council grant B-Q21F
    The second author thanks the Department of Applied Mathematics of the Hong Kong Polytechnic University and especially the first author for the opportunity to visit the department in June 2010 when this project was completed
  • Communicated by: Ken Ono
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 3795-3806
  • MSC (2010): Primary 05C38, 15A15; Secondary 05A15, 15A18
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11232-2
  • MathSciNet review: 2823026