The characteristic polynomial of the given matrix M is x^3 - 3x^2 - 6x + 8. The real eigenvalues of M are -2, 1, and 4. The eigenvectors corresponding to each eigenvalue are: v1 = (11, 6, 3) for λ = -2, v2 = (-2, -3, 3) for λ = 1, and v3 = (1, 0, 0) for λ = 4.