Thus for the first order system of equations there are four independent state variables. When we move to second order form, we will choose to keep the velocities of the masses. For this kind of problem I think it is best to focus on the velocities of the two masses instead of the displacements of the springs. Simply because, in the end, it is the motion of the masses which is of most interest to us. Check, and notice that 3c, 3d, and 3e are not independent.
V is a matrix whose columns are the eigenvectors.
D is a diagonal matrix whose diagonal elements are the eigenvalues. T and I will do so in the following. This solution is valid for any initial conditions. The lack of simple sinusoidal motion showed that the system was not in a pure normal mode of vibration. Let's try one of our new normal modes, to see if our simulation agrees that it is indeed a normal mode.