I know that Real Symmetric Matrices have real eigenvalues and the vectors corresponding to each distinct eigenvalue is orthogonal from this answer. But what if the matrix has repeated eigenvalues? Does it have linearly independent (and orthogonal) eigenvectors? How to prove that?
PS: In the answer I referred to has another answer which might have answered this question. I'm not sure if it answered my question since I didn't understand it. If it did answer my question, can anyone please explain it?