# Can there exist a Non real Hermitian matrix Which is Diagonizable over R

We know any Hermitian matrix is Diagonizable over C always . And If the entries are real then it is a symmetric matrix so it is Diagonizable over R as well as C

So here my Question is.. Does there exist any Hermitian matrix with at least one non real entry such that The matrix is Diagonizable over R

If the hermitian matrix $$A$$ is diagonalizable over the reals, then there are real matrices $$D$$ (diagonal) and $$S$$ (invertible) such that $$A = SDS^{-1}$$. Hence $$A$$ is a product of real matrices. And $$A$$ is real.