I think a system of springs is a good example. I think having a bunch of springs hooked together is a bit abstract so let's instead think of a molecule and model the bonds between the atoms as springs. If you were to squeeze this molecule together or try to pull it apart and then let go, it would vibrate in some complex way. By complex I mean that it wouldn't just bounce back along the direction that you compressed or stretched it.
However, if you write down the matrix of spring constants for the system and solve for the eigenvalues and eigenvectors of this system you can do something special. If you compress or stretch the molecule along the direction of the one of the eigenvectors then let go, the molecule will continue to vibrate along that same direction. The motion will not spread out to all other degrees of freedom. It will also vibrate with a frequency given by the eigenvalue of that eigenvector.
Additionally, any complex vibration of the system can be broken down into a combination of these independent vibrational modes. This is a simple fact because the eigenvectors form an orthogonal basis for the space.
However, if you write down the matrix of spring constants for the system and solve for the eigenvalues and eigenvectors of this system you can do something special. If you compress or stretch the molecule along the direction of the one of the eigenvectors then let go, the molecule will continue to vibrate along that same direction. The motion will not spread out to all other degrees of freedom. It will also vibrate with a frequency given by the eigenvalue of that eigenvector.
Additionally, any complex vibration of the system can be broken down into a combination of these independent vibrational modes. This is a simple fact because the eigenvectors form an orthogonal basis for the space.