Generic Eigenstructures of Hermitian Pencils

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 260-283, March 2024.
Abstract. We obtain the generic complete eigenstructures of complex Hermitian [math] matrix pencils with rank at most [math] (with [math]). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian [math] pencils with the same complete eigenstructure (up to the specific values of the distinct finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases [math], corresponding to general Hermitian pencils, and [math] exhibit surprising differences, since for [math] the generic complete eigenstructures can contain only real eigenvalues, while for [math] they can contain real and nonreal eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures.