Unitary diagonalization of matrices

Question

Can someone tell me whether every square matrix $A\in \mathbb{R}$ unitarily diagonizable? If yes what is a necessary and sufficient condition for a square matrix $A\in \mathbb{R}^{n\times n}$ to be unitarily diagonalizable (please provide justification.)

Answer 1

I assume you mean orthogonally diagonalizable so that you don't leave the real numbers. That is, I assume you ask for which $A \in M_n(\mathbb{R})$ we can find an orthogonal matrix $O$ such that $O^TAO$ is diagonal. This happens if and only if $A$ is symmetric, as guaranteed by the real spectral theorem .