Let $T$ be a self-adjoint linear operator on $H$ = $\Bbb{C}^n$.
Let $x_j$ be a unit eigen vector cooresponding to $\lambda_j$.
I am also given that $T$ has $n$ distinct eigen values.
Define $P: H \rightarrow H$ as
$P(x) = <x,x_j>.x_j$
Then how can I show that $P$ is orthogonal projection of $H$ onto the eigen space of $T$ corresponding to $\lambda_j$?