This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a Hermitian matrix? And if that is the case, could I just find the hermitian matrix and solve for its eigenvalues?. When I thought about it, looks like determining the hermitian matrix from the polynomial equation looks like a daunting task. Any thoughts on this?