Visualising eigenvectors/values

This might seem like an odd question but seeing as I haven't had any formal education in solving ratex models yet, it is something I have been thinking about a lot recently. Consider the following state space representation of an arbitrary model:

$\Gamma_{0}y_{t} = \Gamma_{1}y_{t-1}$

Where the gamma's are coefficient vectors. To solve this system we can simply pre-multiply by $\Gamma_{0}^{-1}$, define $A =\Gamma_{0}^{-1} \Gamma_{1}$ and then decompose $A$ in to its Jordan canonical form $A = P \Lambda P^{-1}$. $\Lambda$ being a vector with eigenvalues on the diagonal and the P's being right and left eigenvectors.

I am fine with applying this method to different models but I feel like I don't have the intuition pinned down. I don't quite understand why I do this decomposition. If I have understood correctly, eigenvalues originate from differential equations where you assume the solution to be on the form $e^{\Lambda t}$ and I suppose that is fine. But then we have to ensure that the eigenvalues are less than one in absolute terms or in some cases that one is inside the unit circle while the other is outside. When solving linear difference equations I can appreciate why we need coefficients to be inside the unit circle (to prevent explosive behavior) but I don't understand how this links to eigenvalues.

So I suppose my question is: what do eigenvector and eigenvalues tell us about the system? Why do we even have to decompose $A$? Is there a visual representation? I am sure in physics eigenvectors have a 'visual' meaning. Is there a similar way to look at eigenvectors in economics? A while back I recall reading a paper that showed the relationship between eigenvectors and the number of initial conditions needed to solve a model. If anyone has references similar to this I would very much like to read them.

I understand that this question is quite vague but I am just trying to understand why I am doing what I am doing.

• More statistics than economics, but I find visualising eigenvectors and eigenvalues almost natural in principal component analysis – Henry Aug 24 '17 at 16:01
• @Henry I am not very familiar with principal component analysis but I'll google it and see if it helps. Thanks. – BenBernke Aug 24 '17 at 20:25