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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:

  1. $V=\mathbb{C}^2$, 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, $$\begin{pmatrix}-1\\0\end{pmatrix},\ \begin{pmatrix}2\\4+4i\end{pmatrix},\ \begin{pmatrix}1+4i\\4i\end{pmatrix},\ \begin{pmatrix}0\\4i\end{pmatrix}.$$
  2. $V=\mathbb{R}[t]_{\leqslant4}$, 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.

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This is less geometric, but might be fun if your students know differential equations. For example, let $E$ be the space of solutions $f$ to $f''-2f'+f=0$. Then the derivative operator $D$ is linear and well defined $E\to E$. (You really need to check this! It's not immediately obvious.) In matrix form $D$ ends up being $\begin{pmatrix}1&1\\0&1\end{pmatrix}$.

In general, you can take any ordinary differential equation with constant coefficients and ask the same question. Now $D$ will map solutions to solutions, since it commutes with the constant coefficient differential operator. Jordan blocks of $D$ in the space of solutions correspond uniquely to roots of the characteristic polynomial, and repeated roots give non-diagonal blocks.

Here the obvious choice of basis (if you know how to solve this kind of differential equations) gives the Jordan normal form. Not much of an exercise there, but the more important lesson is in checking that $D$ indeed maps $E\to E$. Checking that your operator is well defined and linear between the spaces you want is a good thing to do. And it might come as a surprise that in this relatively natural setting (space of solutions and the derivative operator) you always get non-diagonal Jordan blocks.

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  1. See the following.

    A. Bujosa, R. Criado, C. Vega. Jordan Normal Forms via Elementary Transformations. Siam Review, Vol. 40, n° 4, p. 947-956.1998.

If we know the eigenvalues of $A$, then in the above and pretty method, we perform only linear combinations of rows and columns to obtain the Jordan form of $A$.

  1. Now we choose randomly $A\in M_n$: then generically, the eigenvalues of $A$ are distinct and $A$ is diagonalizable (over $\mathbb{C}$). Now we choose randomly a linear application with at least a multiple eigenvalue; then $A$ is "never " diagonalizable. For example, let $n=2$ and $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ be s.t. it has only the eigenvalue $1$, that is, $A=\begin{pmatrix}a&b\\\dfrac{2a-a^2-1}{b}&2-a\end{pmatrix}$ or $A=\begin{pmatrix}1&0\\c&1\end{pmatrix}$; generically, $A$ is never $I_2$.

We assume $n=5$ and that $A$ has spectrum $0,0,0,1,1$ and we randomly choose such a matrix $A$. We can simulate the random choice of $A$ as follows (so that $A\in M_5(\mathbb{Z})$):

Randomly choose

i) $T=diag(U_3,V_2)$ where $U_3,V_2$ are upper-triangular and with diagonals $[0,0,0],[1,1]$.

ii) $P=LU$ where $L$ is lower and $U$ upper triangular and s.t. the entries of the diagonals are $\pm 1$.

and put $A=P^{-1}TP$. Then "always", $A$ has $diag(J_3,I+J_2)$ as Jordan form (where $J_k$ is the nilpotent Jordan block of dimension $k$)!!

Note that $I+J_2$ is a shear in the plane and that $J_k=Q\pi$ where $Q=[q_{i,j}]$ is the permutation matrix defined by $q_{i,j}=0$ except $q_{i,i+1}=1,q_{n,1}=1$ and $\pi$ is the projection $diag(0,1,1,\cdots)$.

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