Question: What is the maximum number of pairwise non-similar linear transformations on a three-dimensional vector space, each of which has the characteristic polynomial $x^3$?
So far I have managed to get some non-similar transformations but I do not know how to use Jordan Form (this question is related to the Jordan From) and do not know how to proceed actaully. Thanks in advance.
If your matrix $A$ has characteristic polynomial $x^3$, then it has only one eigenvalue, i.e. $0$ with algebraic multiplicity $3$.
There are three possibilities, depending on the geometric multiplicity of $0$. If it's $3$, then your matrix is similar to $$\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right] $$
If it's $2$, then your matrix is similar to $$\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} \right] $$
If it's $1$, then your matrix is similar to $$\left[ \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} \right] $$ And there are no other possibilities.