Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence

We construct the Hasse diagrams G2G2 and G3G3 for the closure ordering on the sets of congruence classes of 2×22×2 and 3×33×3 complex matrices. In other words, we construct two directed graphs whose vertices are 2×22×2 or, respectively, 3×33×3 canonical matrices under congruence, and there is a directed path from A to B if and only if A can be transformed by an arbitrarily small perturbation to a matrix that is congruent to B.A bundle of matrices under congruence is defined as a set of square matrices A   for which the pencils A+λATA+λAT belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of 2×22×2 or 3×33×3 matrices have the same properties with respect to perturbations. We construct the Hasse diagrams G2B and G3B for the closure ordering on the sets of congruence bundles of 2×22×2 and, respectively, 3×33×3 matrices. We find the isometry groups of 2×22×2 and 3×33×3 congruence canonical matrices.