The vibration of a bell (say) could be modelled by a matrix, with a state vector to represent position, velocity, and acceleration, and the matrix modelling the differential equations describing their evolution over time. The eigenvectors represent a basis of the system, so that we can describe any potential state vector as a sum of eigenvectors. If we do so, then each step of the system can be modelled by multiplying each of these eigenvectors by its corresponding eigenvalue. If an eigenvalue happens to be complex, then we can describe it in phasor form as the product of an amplitude and a angle. The amplitude tells us how it will decay (or amplify) over time. The angle tells us the frequency of oscillation, and thus the note that the bell will typically sound.