You have seen how matrices and eigenvectors can be central to the analysis of moving bodies. Now that your attention is on vibration, you will find that they can be central to that too! There are some fundamental features of vibration: As an engineer, you will probably wish to add some damping to limit any unwanted vibration or to change the ‘resonant’ frequency away from that of a disturbance. As a musician, you would wish to shape the vibration to something that sounds pleasant! So to deal with a vibrating system you should: Now there are two ways to go about step 3. You can mess about with simultaneous equations and algebra, or you can use the power of matrices to help you: This gives an exponential solution that is easy to recognise and analyse.