Change of the *-congruence canonical form of 2-by-2 matrices under perturbations

Abstract

It is constructed the Hasse diagram for the closure ordering on the sets of *congruence classes of 2 × 2 matrices. In other words, it is constructed the directed graph whose vertices are 2 × 2 canonical complex matrices for *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.