I know that definition of orthogonal projection matrix is its range and null space are orthogonal.
And, definition of projection matrix is $P=P^2$
Then, I understand that if $P$ is $n\times n$ projection matrix. $P=P^T \Rightarrow P$ is orthogonal projection matrix(using fundamental theorem of linear algebra)
Then, how could I proof that $P$ is orthogonal projection matrix $\Rightarrow P=P^T$
Can I solve it with using SVD?
For any projection $P$, the kernel of $P$ is the image of $I-P$, and $\ker P \oplus \Im P = V$. So you only need to show that if these two spaces are orthogonal, then $P$ is self adjoint. We can directly check: $$\langle Px, y\rangle = \langle Px, (I-P)y + Py\rangle = \langle Px, Py \rangle, $$ and similarly $\langle x, Py \rangle = \langle Px, Py \rangle $, so we have equality.