It probably isn’t surprising that an exact solution can be found for eigenvalues of a real 3×3 symmetric matrix. This conclusion follows from noting that the characteristic equation is cubic, for which an exact solution procedure can be found in any good algebra reference. One thing that you won’t find in many such resources, however, is an algorithm for the solution that will avoid complex numbers in the intermediate steps of the calculation whenever the components of the source symmetric 3×3 matrix are all real.