Generic symmetric matrix pencils with bounded rank
Abstract
We show that the set of $n \times n$ complex symmetric matrix pencils of rank at most $r$ is the union of the closures of $\lfloor r/2\rfloor +1$ sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of $n \times n$ complex symmetric matrix pencils of rank at most $r$. We also show that these closures correspond to the irreducible components of the set of $n\times n$ symmetric matrix pencils with rank at most $r$ when considered as an algebraic set.
 Publication:

arXiv eprints
 Pub Date:
 August 2018
 arXiv:
 arXiv:1808.03118
 Bibcode:
 2018arXiv180803118D
 Keywords:

 Mathematics  Spectral Theory;
 15A22;
 15A18;
 15A21;
 65F15
 EPrint:
 15 pages