Real eigenvalues, similar symmetric matrix

Question

I know that symmetric matrices have real eigenvalues, and that non-symmetric matrices that are similar to symmetric matrices must also have real eigenvalues, but is the converse true?

That is, if a matrix has real eigenvalues, must there exist a similar matrix that is symmetric?

Answer 1

No. Counterexample: $$ \pmatrix{0&1\\0&0} $$ is not similar to any symmetric matrix.

On the other hand, every diagonalizable matrix with real eigenvalues is similar to a symmetric matrix.