Does symmetry of a matrix imply orthogonally diagonalizable?

Question

If you have a symmetric matrix, is it orthogonally diagonalizable? Or is the converse only true?

Answer 1

They are equivalent ,

$A$ being orthogonally diagnoizable you mean that there's an orthogonal matrix $U$ and a diagnonal matrix $D$ such that $A=UDU^{−1}=UDU^T$.

$A$ is then symmetric,( since $D$ is diagnonal, $D^T=D$)

$$A^T=(UDU^T)^T=(DU^T)^TU^T=UD^TU^T=UDU^T=A.$$