Abstracts

Removing critical eigenvalues of Hermitian matrix pencils and Hamiltonian matrices with minimal perturbations

Shreemayee Bora,  IIT Guwahati

It is well known that eigenvalues of a Hermitian matrix pencil $L(\lambda) = \lambda A - B$, occur in pairs $(\lambda, \overline{\lambda})$ with eigenvalues, if any, on the extended real line being the ones where the symmetry breaks down. We refer to such eigenvalues as critical eigenvalues. When $L(\lambda)$ is of even size, we study the problem of finding the smallest Hermitian perturbations of $L(\lambda)$ such that the perturbed Hermitian matrix pencil is regular and all its critical eigenvalues, if any, can be removed by a further arbitrarily small Hermitian perturbation. Additionally, the minimal perturbations are required to be real if $L(\lambda)$ is real. We find an algorithm for estimating the distance to a nearest regular Hermitian matrix pencil without critical eigenvalues. It is observed that when $L(\lambda)$ is randomly generated, the algorithm provides either theexact distance or a very good estimate of it.

Under additional assumptions on the coefficient matrix $A$ of $L(\lambda)$, we also study the same problem while perturbing only $B$. In such cases, we show that the distance can be exactly computed if $L(\lambda)$ is a definite matrix pencil. However, in practice we find that our algorithm computes the exact distance in other cases as well. The same algorithm also estimates the distance to bounded realness for Hamiltonian matrices. This distance can be exactly computed in those cases where the Hamiltonian matrix has only purely imaginary eigenvalues with the ones of positive type being separated from those of negative type. However, as experiments demonstrate, this can also happen when the Hamiltonian matrix does not satisfy this condition.