Consider the matrix A=uv* where u and v lie in C^n (C:complex numbers). Under what condition on u and v is A a projector?

I know that a projector is a square matrix P for which P^2= P.

Now according to this definition, would that make uv*=Identity. and then u be the inverse of v*?

Also what is the inverse of a vector that belongs to C^n? I mean what is u^-1? Is u^-1 a vector such that uu^-1 = the (nxn) identity matrix ?