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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


Spectral decomposition of real symmetric quadratic $ \lambda$-matrices and its applications

Authors: Moody T. Chu and Shu-Fang Xu
Journal: Math. Comp. 78 (2009), 293-313
MSC (2000): Primary 65F15, 15A22, 65F18, 93B55
Published electronically: June 24, 2008
MathSciNet review: 2448708
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Abstract: Spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions. The canonical form often facilitates discussions which, otherwise, would be complicated and involved. Spectral decomposition is of fundamental importance in many applications. The well-known GLR theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees, but its development is based on complex numbers. This paper modifies the GLR theory for the special application to real symmetric quadratic matrix polynomials, $ \mathcal{Q}(\lambda)=M \lambda^{2} + C \lambda + K$, $ M$ nonsingular, subject to the specific restriction that all matrices in the representation be real-valued. It is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which, in turn, can be constructed from the spectral data. Applications to a variety of challenging inverse problems are discussed.

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Additional Information

Moody T. Chu
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Shu-Fang Xu
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

PII: S 0025-5718(08)02128-5
Keywords: Quadratic $\lambda $-matrix, spectral decomposition, real standard pair, inverse eigenvalue problem
Received by editor(s): March 28, 2007
Received by editor(s) in revised form: December 12, 2007
Published electronically: June 24, 2008
Additional Notes: Research of the first author was supported in part by the National Science Foundation under grants DMS-0505880 and CCF-0732299.
Research of the second author was supported in part by NSFC under grant 10571007.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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