Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils
Andrii Dmytryshyn, Bo Kågström · SIAM Journal on Matrix Analysis and Applications · 2014
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. The developed 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 $C-\lambda D$.