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2006.175: The nearest definite pair for the Hermitian generalized eigenvalue problem

2006.175: Sheung Hun Cheng and Nicholas J. Higham (1999) The nearest definite pair for the Hermitian generalized eigenvalue problem. Linear Algebra and its Applications, 302-303. pp. 63-76. ISSN 0024-3795

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DOI: 10.1016/S0024-3795(99)00026-9


The generalized eigenvalue problem $Ax = \lambda Bx$ has special properties when $(A,B)$ is a Hermitian and definite pair. Given a general Hermitian pair $(A,B)$ it is of interest to find the nearest definite pair having a specified Crawford number $\delta > 0$. We solve the problem in terms of the inner numerical radius associated with the field of values of $A+iB$. We show that once the problem has been solved it is trivial to rotate the perturbed pair $(A+\dA,B+\dB)$ to a pair $(\widetilde{A},\widetilde{B})$ for which $\lambda_{\min}(\widetilde{B})$ achieves its maximum value $\delta$, which is a numerically desirable property when solving the eigenvalue problem by methods that convert to a standard eigenvalue problem by ``inverting $B$''. Numerical examples are given to illustrate the analysis.

Item Type:Article
Uncontrolled Keywords:Nearest definite pair; Crawford number; Hermitian pair; Generalized eigenvalue problem; Field of values; Inner numerical radius; Numerical radius
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2006.175
Deposited By:Miss Louise Stait
Deposited On:07 July 2006

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