Usually it is something like the eigenvectors represent stable states of the system, and other states will tend to be unstable until and decay into one of those stable states.

For example, the eigenvectors of the moment of inertia tensor represent “principle axes” of rotation, and these represent the possible stable axes of rotation (usually only one or two axes is actually stable, it depends on the object).

By analyzing principle axes of inertia, you can explain why a frisbee’s rotation is very stable around one axis but unstable around all other axes. And you can predict this kind of behavior for other objects.

Another example is in quantum mechanics, eigenvectors correspond to states that result after “measurement collapse” of the wavefunction, and are useful in various quantum mechanics problems, such as predicting the behavior of atoms, molecules, or semiconductors.

The largest eigenvector would be the most probable direction and velocity of the struck object after impact?

The size of the eigenvector doesn’t really matter, because if a vector is an eigenvector, scaling it (changing its length without changing the direction) will also result in an eigenvector. It’s the direction of the eigenvector that matters.

However, the eigenvalue does matter and often has real world implications, for example, it can help you determine which of the principle axes of rotation will result in a stable rotation .

An eigenvector doesn’t change direction when it is multiplied by the matrix, but it might change its length. The amount that length changes is the eigenvalue. vM=ev where M is a matrix, v is an eigenvector of M, and e is the corresponding eigen value.