Suppose that $\{u_1, \dots, u_r\}$ is an orthonormal set in $\Bbb C^n$ ($r \leq n$), and consider the matrix $R$ whose columns are these vectors: $R = [u_1, \dots, u_r].$ Show that $R^\ast R$ and $RR^\ast$ are projectors, one of them being $I$.
What I did is this: Since $u_1, \dots, u_r$ are orthonormal vectors in $\Bbb C^n$, then:
$(R^\ast R)^2 = (R^\ast R)(R^\ast R) = R^\ast RR^\ast R = R^\ast (RR^\ast)R = R^\ast(I)R = R^\ast R$ and $(RR^\ast)^2 = (RR^\ast)(RR^\ast) = RR^\ast RR^\ast = R(R^\ast R)R^\ast = R(I)R^\ast = RR^\ast.$
Is this right? But how can I show that one of them is $I$? Well, in my opinion, I may think both of them are $I$.