I wanted to ask this question since I have seen conflicting viewpoints on it.
Are orthogonal matrices necessarily symmetric? I do not believe so but some website said they were so I need to confirm. (I think they are only so if they equal their transpose).
Secondly, are orthogonal matrices necessarily diagonalizable? If they are symmetric, they are, but what about if they are not? Are they still always diagonalizable?
They are not necessarily symmetric: $$ \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}. $$ The above is also not diagonalizable over $\mathbb{R}$. Orthogonal matrices are however always diagonalizable over $\mathbb{C}$-this follows from the spectral theorem.