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New proofs and extensions of Sylvester’s and Johnson’s inertia theorems to non-Hermitian matrices

Authors: Man Kam Kwong and Anton Zettl
Journal: Proc. Amer. Math. Soc. 139 (2011), 3795-3806
MSC (2010): Primary 05C38, 15A15; Secondary 05A15, 15A18
Published electronically: June 28, 2011
MathSciNet review: 2823026
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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

Anton Zettl
Affiliation: Department of Mathematics, Northern Illinois University, DeKalb, Illinois 60115

Keywords: Matrix eigenvalues, positive and negative eigenvalues, eigenvalue curves
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.