A question on one of my past exam papers is
1. List all possible Jordan normal forms of nilpotent 4 x 4 matrices
How would you go about doing this?
The char. polynomial of a nilpotent 4x4 matrix is $\displaystyle x^4$ , the question now is: what's its minimal polynomial$\displaystyle m(x)$?
For example, if $\displaystyle m(x)=x^2\Longrightarrow $ there's at least one Jordan Block (JB) of order 2, so you have two possibilities for the JNF of a matrix like this:
either $\displaystyle \begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\ end{pmatrix}$ --- one 2x2 JB + two 1x1 JB's , or
$\displaystyle \begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\ end{pmatrix}$ --- two 2x2 JB's.
Well, now check the other possibilities for the minimal polynomial.
Tonio