Eigenvalue Theory
A mode of vibration is a characteristic way in which an object can vibrate. The number of modes depends on the eigenvalues. So if an object or system has three Eigenvalues then it has three modes of vibration. This means that the bridge could vibrate up, sideways, or possibly a compression direction. Finding the eigenvectors for structures allows engineers to create designs that can withstand the vibrations. The Eigenvalues can be put in a limit formula to determine the stability of an object. For example, if the limits come out negative then the object might will be stable, if they come out positive then the object will not be. Of course every situation is unique and no golden rule can be applied.