Miniversal deformations of pairs of symmetric forms
We give a miniversal deformation of each pair of symmetric matrices $(A,B)$ under congruence; that is, a normal form with minimal number of independent parameters to which all matrices $(A+E,B+E')$ close to $(A,B)$ can be reduced by congruence transformations $ (A+E,B+E')\mapsto {\cal S}(E,E')^T (A+E,B+E') {\cal S}(E,E'), {\cal S}(0,0)=I, $ in which ${\cal S}(E,E')$ smoothly depends on the entries of $E$ and $E'$.