# Does symmetry of a matrix imply orthogonally diagonalizable? [duplicate]

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

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