Assume you have a class of students more or less familiar with the notion of the matrix of a linear operator. They have seen and calculated lots of examples in various context: geometric transformations (rotations, reflections, scaling along axes, ...), operators on polynomials (derivation), number-theoretic ($\mathbb{C}^n\to\mathbb{C}^n$, linear over $\mathbb{R}$ but not over $\mathbb{C}$).

In the study of the Jordan normal form the basic problem is to find the canonical form and a Jordan basis of an operator. The algorithm one usually gives to the students starts with the line “pick a basis and find the matrix of the given operator with respect to this basis”. But then we give the students a problem of the form “given *a matrix*, find its canonical form and a Jordan basis”.

Now I would very much like to force the students to calculate the Jordan form of *an operator*, so they would pick a basis themselves, find the corresponding matrix, find the Jordan basis and then express it not as a set of columns of numbers, but as elements of the vector space in question.

This needs a couple of examples, here they are:

- $V=\mathbb{C}^2$, the operator is $A\colon\begin{pmatrix}x\\y\end{pmatrix}\mapsto\begin{pmatrix}\overline{x}-\operatorname{Re}(y)\\(1+i)\cdot\operatorname{Im}(x)-y\end{pmatrix}$. The natural $\mathbb{R}$-basis is $$\begin{pmatrix}1\\0\end{pmatrix},\ \begin{pmatrix}i\\0\end{pmatrix},\ \begin{pmatrix}0\\1\end{pmatrix},\ \begin{pmatrix}0\\i\end{pmatrix},$$ the matrix of $A$ is $$\begin{pmatrix}1&0&-1&0\\0&-1&0&0\\0&1&-1&0\\0&1&0&-1\end{pmatrix},$$ the JNF is $\operatorname{diag}(1,J_2(-1),-1)$, and the Jordan basis is, for example, $$\begin{pmatrix}-1\\0\end{pmatrix},\ \begin{pmatrix}2\\4+4i\end{pmatrix},\ \begin{pmatrix}1+4i\\4i\end{pmatrix},\ \begin{pmatrix}0\\4i\end{pmatrix}.$$
- $V=\mathbb{R}[t]_{\leqslant4}$, the space of polynomials of degree at most 4, and the operator if $f\mapsto f'+f(0)+f'(0)$. The Jordan basis in this case is a set of polynomials.

The two examples above are not very interesting in terms of the calculating the JNF (few small blocks, distinct eigenvalues), but this can be easily fixed.

But I find it pretty hard to invent a problem of this sort which have a geometric origin (transformations in, say 4- or 5-dimensional Euclidean space). Most of the transformations I can describe in simple geometric terms (rotations, reflections, projections) are either diagonalizable, or have imaginary eigenvalues (so it is impossible to get back form the coordinate columns to points in space), or both.

Is there a way to construct a “geometric” problem on the computation of the JNF?

Since there must be other contexts similar to the three described above,

what are the interesting problems on the computation of the JNF of a particular operator?

To clarify this second question, I am well-aware of the problems of the sort “one knows the characteristic and minimal polynomials, the rank of the square and the maximal number of linearly independent eigenvectors, find the JNF”.

Apart from the use in the class in order for the students to recall the notion of the matrix of an operator, this can also be very useful in an online course with automated assignment check.