Let $A \in \mathbb{C}^{n \times n}$ be a diagonalizable matrix and let $\mathfrak{A_1}, \dots \mathfrak{A}_n$ the eigenspaces such that
$$\mathbb{C}^{n} = \mathfrak{A_1}\oplus \dots \oplus\mathfrak{A_n}$$
So One can define the projection on each eigenspace by the map $P_i:\mathbb{C}^n \rightarrow \mathfrak{A_i}$ by $Px=x$ if $x \in \mathfrak{A_i}$ and $Px=0$ otherwise.
1)Is this definition correct?
2)Are the projections always mutually orthogonal assuming these hypotheses? I think this immediately follows from the fact that I'm writing the original space as a direct sum.