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Detecting hyperbolic and definite matrix polynomials

✍ Scribed by V. Niendorf; H. Voss

Book ID
104038122
Publisher
Elsevier Science
Year
2010
Tongue
English
Weight
379 KB
Volume
432
Category
Article
ISSN
0024-3795

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✦ Synopsis

Hyperbolic or more generally definite matrix polynomials are important classes of Hermitian matrix polynomials. They allow for a definite linearization and can therefore be solved by a standard algorithm for Hermitian matrices. They have only real eigenvalues which can be characterized as minmax and maxmin values of Rayleigh functionals, but there is no easy way to test if a given polynomial is hyperbolic or definite or not. Taking advantage of the safeguarded iteration which converges globally and monotonically to extreme eigenvalues we obtain an efficient algorithm that identifies hyperbolic or definite polynomials and enables the transformation to an equivalent definite linear pencil. Numerical examples demonstrate the efficiency of the approach.

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In this paper we study the space L p ( +), 1 p + , + being a positive definite matrix of measures. We prove that the set of all positive definite matrices of measures having the same moments as those of + is compact in the vague topology, and we give a density result for L 1 ( +) which is an extensi