The nearest definite pair for the Hermitian generalized eigenvalue problem

AbstractThe generalized eigenvalue problem Ax=λBx 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 δ>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+ΔA,B+ΔB) to a pair (Ã,B̃) for which λmin(B) achieves its maximum value δ, 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