Miniversal deformations of pairs of skew-symmetric matrices under congruence

Abstract

Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair $(A,B)$ we provide a normal form with a minimal number of independent parameters to which all pairs of skew-symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation which smoothly depends on the entries of the matrices in the pair $(\widetilde{A},\widetilde{B})$. An upper bound on the distance from such a miniversal deformation to $(A,B)$ is derived too. We also present an example of using miniversal deformations for analyzing changes in the canonical structure information (i.e. eigenvalues and minimal indices) of skew-symmetric matrix pairs under perturbations.