Given characteristic polynomial $(x-3)^3 (x+1)$ and minimal polynomial $(x-3)^2 (x+1)$. Find possible Jordan forms of linear transformations with them.
Here is my answer:
\begin{bmatrix} 3 & 1 & 0 &0 \\ 0 & 3 & 0 &0 \\ 0&0&3&0\\ 0&0&0&-1 \end{bmatrix}
My questions are:
1), Does it matter if I start with $3$ or $-1$ on the left (i.e. does the magnitude of eigenvalues matter in Jordan forms)?
2), Does it matter if I put "$1$" at the entry(1,2) (i.e. first row and second column) or at the entry(2,3) (i.e. second row and third column)? In the above matrix, I put "$1$" at the entry(1,2).
No, it does matter in either case, but that means that you need to count the permutations of the Jordan blocks. Thus, there are 3! = 6 possible Jordan forms.