Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skewsymmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another skew-symmetric matrix pencil. The developed theory relies on our main theorem stating that a skew-symmetric matrix pencil A− λB can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C − λD if and only if A − λB can be approximated by pencils congruent to C − λD. The stratification theory is also illustrated by using StratiGraph.