Groups and semigroups generated by a single unitary orbit

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given invertible matrix A. In particular, we give necessary and sufficient conditions for such a semigroup to be the special linear group or the general linear group or other classical matrix groups. Furthermore, we use the above to determine the subgroups of the general linear group that are normalized by the unitary group.