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# 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 skew-symmetric 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. This theory relies on our main theorem stating that a skew-symmetric matrix pencil A-\lambda B can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C-\lambda D if and only if A-\lambda B can be approximated by pencils congruent to C-\lambda D.

### Keywords

Skew-symmetric matrix pencil; Stratification; Canonical structure information; Orbits

### Authors

Andrii Dmytryshyn and Bo Kågström

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Page Responsible: Frank Drewes
2023-03-30