My instructor explicitly stated that, because we are asked to find eigenvalues and eigenvectors of a $7\times 7$ matrix, MATLAB would be easiest to use. The equation $(1)$ is intended to resemble a mass-on-spring system, so I'm aware of how to find the natural frequency of the system, and I'm also aware that it's undamped. In other words, I can interpret and solve the equation to an extent.
What I'm specifically struggling with is writing the model as a second order system and, as a result, finding the eigenvalues and eigenvectors of the system. Any coding help, or for that matter any help at all, would be greatly appreciated.