Congruential automorphism groups of general matrices

For , Aut(A), the congruential automorphism group of A, is the set of all non-singular such that C*AC=A. Here, the A for which Aut(A) is bounded are described, and this allows the characterization of pairs A, B for which the set of matrices carrying the congruence of A to B is compact. This characterization is in terms of non-singular unitoid matrices in which certain configurations of canonical angles are forbidden.