Question states: "Suppose $A$ is symmetric matrix. Show that the following are equivalent:
(1) $A$ is orthogonal.
(3) All eigenvalues of $A$ are $\pm 1$"
I proved that 1 and 2 are equivalent, and that 1 or 2 implies 3. It remains to prove that 3 implies either 1 or 2.
Here is my proof: since $A$ is symmetric, it is orthogonally diagonalizable. We'll work in the basis such that $A$ diagonal. Then $A$ has all its diagonals as $\pm 1$ and all its nondiagonal entries as $0$, and so $A$ is orthogonal. Does this prove make sense? Am I really safe to assume without loss of generality that $A$ is diagonal? If I am wrong, could you provide an explanation for why 3 implies 1 or 2?