Sigma-critical representations are quiver representations that satisfy certain matrix equations. They arise naturally in the context of the Kempf-Ness theorem on closed orbits in Invariant Theory. After introducing all the relevant concepts, I will first describe a result that gives necessary and sufficient conditions for the orbit of a representation to contain a sigma-critical representation. I will then explain how this result can be used to solve the Paulsen Problem for matrix frames. This is based on joint work with Jasim Ismaeel.