Eigen Decomposition of Linear Systems

One of the basic tools for understanding linear, time-invariant systems is the 'modal' decomposition on to a set of 'eigendirections'.  This is illustrated in the matlab files in the Eigen Decomposition Demo download.

Run the script S_EigenDemo.m to create versions of the plot shown above. The script selects random eigenvectors and assigns eigenvalues for a 3D system, then calls a function to superimpose a trajectory over a representation of the eigendirections.

To test your understanding, try the following exercises:

1. Change the real parts of the eigenvalues to obtain other types of behavior.  What happens if you make the real eigenvalue much more negative, or slightly positive? What happens if you make the real part of the complex conjugate pair slightly positive?
2. How can you make the simulated trajectory orbit the real eigenvector in a plane parallel to the other two eigendirections?