It is well know that over algebraically closed field field two matrices are similar if and only if they have same Jordan normal form(up to permutation of blocks).
So undergraduates spend a lot of time mastering this topic.
It is interesting by its own right, just from purely linear algebra viewpoint. However, I think the most useful way to show why it is interesting outside linear algebra is to provide examples showing classification of misc. mathematical objects from other branches of mathematics. Or, maybe, situations when JNF allows to prove something that is cool or exciting. Maybe something not evident, catchy. Maybe something from research papers, or from memorable expository papers.
So, can you please share your favourite applications of Jordan normal form in mathematics? Thanks a lot for your time and contribution!
Update: I believe my questions is different from What is the purpose of Jordan Canonical Form? because the main things I am interested in is how Jordan normal form is used in real life research. So ideal answer is the following: In my paper X I used Jordan normal from to prove Y or classify Z. I have perspective about motivation of Jordan normal from, I want to know more about it application in real-life research or illuminating exposition.