I learned below theorem and there is a proof that orthogonally diagonalizable matrix is symmetric, but there is no proof that symmetric matrix are orthogonally diagonalizable.
Theorem 2. An $n\times n$ matrix $A$ is orthogonally diagonalizable if and only if $A$ is a symmetric matrix.
I searched proof in this website and found this proof, but I cannot understand why $\langle Ax,y \rangle = \langle x, A^Ty \rangle$.
How can I prove?
Writing the inner product $\langle v, w \rangle$ as $v^\top w$, we have $$\langle Ax, y\rangle = (Ax)^\top y = x^\top A^\top y = \langle x, A^\top y\rangle.$$