Hint: Real symmetric matrices are (orthogonally) diagonalisable. And all eigenvalues of nilpotent matrices are zero.

Is nilpotent matrix zero?

A square matrix A is called nilpotent if some power of A is the zero matrix. Namely, A is nilpotent if there exists a positive integer k such that Ak=O, where O is the zero matrix.

How do you know if a matrix is nilpotent?

An n×n matrix A is called nilpotent if Ak=O, where O is the n×n zero matrix. Prove the followings. (a) The matrix A is nilpotent if and only if all the eigenvalues of A is zero. (b) The matrix A is nilpotent if and only if An=O.