# The nearest definite pair for the Hermitian generalized eigenvalue problem

Cheng, Sheung Hun and Higham, Nicholas J. (1999) The nearest definite pair for the Hermitian generalized eigenvalue problem. Linear Algebra and its Applications, 302-30. pp. 63-76. ISSN 0024-3795

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.