TITLE: What is the purpose of Jordan Canonical Form?
QUESTION [18 upvotes]: I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. 
What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler, but I don't claim to be an expert) and have taken abstract algebra (most familiar with group and ring theory) and have briefly skimmed through linear algebra books covering this material, but I don't quite understand the "big picture" idea, i.e., why is this useful in application? One person once told me it is the "most straightforward and useful algorithm for solving systems of linear equations, once you get beyond 3 variables or so," but maybe I'm missing something, since I usually don't see anything like what this person described to me in the linear algebra books I have. Most textbooks I've seen tend to have a more theoretical focus on this topic.
Also, any suggested texts which have good coverage on this topic would be very helpful.

REPLY [5 votes]: Among other things the Jordan form can show that any Markov process must terminate and also helps to find the limiting state. Instructors usually motivate such processes by asking how to find equilibrium states in chemistry, but this also has applications to google's page rank system. Basically rank pages due to the largest entry in the limiting state.