# Motivational examples of using Jordan normal form in mathematics.

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.

• The answers at this question may be helpful. – Mark S. May 10 '17 at 15:52
• @Moo I wouldn't say that this present question is a duplicate of the question you mention (I have had also a look at the answers which are not truly in the spirit of finding exciting aspects for student's motivation); thus, I don't agree to close this question. – Jean Marie May 10 '17 at 16:01

It is fundamental in dynamical systems. The most general constant coefficient linear ordinary differential equation is $$\frac{d}{dt} u(t)= Au(t),$$ where $$A$$ is a $$n\times n$$ matrix and $$u\in \mathbb R^n$$. The solution is written as $$u(t)=e^{At} u_0, \quad \text{where }u_0=u(0),$$ and where the matrix exponential is $$e^{At}=\sum_{m=0}^\infty \frac{t^m A^m}{m!}.$$ The Jordan normal form gives a way to compute $$e^{At}$$; up to a linear change of coordinates it is a block matrix, with triangular blocks having $$e^{\lambda t}$$ on the diagonal. Here, $$\lambda$$ are the eigenvalues of $$A$$.

The conclusion is that the long-time dynamics, that is, the limit as $$t\to \infty$$, is entirely driven by the eigenvalues of $$A$$. The directions corresponding to Jordan blocks with $$\Re \lambda >0$$ will be exponentially diverging, the ones with $$\Re \lambda <0$$ will be exponentially converging to the origin, the other ones will need more treatment.

This is huge, and vastly used in nonlinear systems as well, via the concept of linearization. This Wikipedia contains some simple explanations.

As discussed in Wikipedia article "Jordan normal form," important results that are consequences include the spectral mapping theorem, Cayley-Hamilton theorem, minimal polynomial, and invariant subspace decompositions.

An example I came across recently.

I like to express recursive sequences using matrices which provide the step of -for instance- $$[a_{k-1},a_k]$$ to $$[a_k,a_{k+1}]$$ by one matrix-multiplication with a matrix $$M$$ and for $$n$$ steps by the $$n$$'th power of $$M$$.
Then using diagonalization one can even determine fractional iterates, and a fairly common example is the Binet-type formula for sequences of the Fibonacci-type which occurs very naturally by the diagonalization-mechanism.
Another view on the same thing is, that one can write formulae for the $$n$$'th iterate keeping $$n$$ indeterminate, a common exercise in learning recursive sequences.
A recent example is at this link asking for $$f(n) = 2f(n-1) -f(n-2) + 1 ,\qquad \qquad f(0)=a, \qquad \qquad f(1)=b \tag 1$$ (As a new(?) exercise, there is even a very recent duplicate of the problem at this link)

However, there are recursions, where the transfermatrix $$M$$ can not be diagonalized. Just recently I came to such a question here in MSE with the recursion $$\ a_n = 3a_{n-1} - 2a_{n-2} + 1 , \qquad\qquad \ a_0 = 2, \qquad\qquad\ a_1 = 3 \tag 2$$ Here the Jordan-decomposition comes into play. It provides the analoguous way to arrive at fractional iterates and closed expressions keeping the iteration-parameter $$n$$ indeterminate in a very similar way as in the diagonalizable cases.

As I find this matrix-approach an amazing approach in general so might do some students as well.

(I did not yet compose an answer to the linked problem because I'm myself still exploring the best procedere and shall give a solution only later.)