Suppose that $U_n(\mathbb{C})$ is the group of unitary matrices of dimension $n$ over complex numbers. Fix a unitary matrix $A \in U_n(\mathbb{C})$ and consider the smallest subgroup $K \subseteq U_n(\mathbb{C}) $, that contains all diagonal matrices (maximal torus group) and also $A$. It seems that apart from some exceptions for $A$ (like when $A$ is a diagonal matrix), $K = U_n(\mathbb{C}) $. Do you have any idea, if this is true, how to prove it and how to derive the exception cases?

Particularly, I'm interested in the case where $A$ is a Circulant matrix, i.e. it has the following form: $$A=F^{-1}\cdot L\cdot F,$$ where $L$ is a diagonal matrix and $F$ is the DFT matrix. I'm not sure if this restriction simplifies the problem or not.

P.S. I have asked the question here in math.stackexchange group, but I guess, the question should be more relevant to this group.