Can the projective unitary group be realized as a matrix Lie group

Question

Is the projective unitary group $U(n)/U(1)$ isomorphic to some closed subgroup of the $GL(n,\mathbb{C})$ (i.e., a matrix Lie group)?

Answer 1

The standard way: Take the Lie algebra of that group. It is finite dimensional. The group acts on it (adjoint action). This gives a representation.