Generic eigenstructures of Hermitian pencils

09/21/2022

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by   Fernando De Terán, et al.

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We obtain the generic complete eigenstructures of complex Hermitian n× n matrix pencils with rank at most r (with r≤ n). 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 n× n pencils with the same complete eigenstructure (up to the specific values of the finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases r=n, corresponding to general Hermitian pencils, and r<n exhibit surprising differences, since for r<n the generic complete eigenstructures can contain only real eigenvalues, while for r=n they can contain real and non-real eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures of Hermitian pencils.