How many possible Jordan decompasitions for an Endomorphism?

Question

Let V be an vector space and $\phi \in End_\mathbb{C}(V)$ a linear map with the characteristic polynomial $P_\phi(X) = (x-2)^3(x-5)^2$. How many possible Jordan normal form are there for $\phi$?

Answer 1

Hint: You should also provide the minimal polynomial that could be one of

• $(x-2)(x-5)$,

• $(x-2)(x-5)^2$,

• $(x-2)^2(x-5)$,

• $(x-2)^2(x-5)^2$,

• $(x-2)^3(x-5)$,

• $(x-2)^3(x-5)^2$.

With that in mind the posibles respectively are

$\left[\begin{array}{ccccc} 2&0&0&0&0\\ 0&2&0&0&0\\ 0&0&2&0&0\\ 0&0&0&5&0\\ 0&0&0&0&5\\ \end{array} \right]$ , $\left[\begin{array}{ccccc} 2&0&0&0&0\\ 0&2&0&0&0\\ 0&0&2&0&0\\ 0&0&0&5&1\\ 0&0&0&0&5\\ \end{array} \right]$ , $\left[\begin{array}{ccccc} 2&1&0&0&0\\ 0&2&0&0&0\\ 0&0&2&0&0\\ 0&0&0&5&0\\ 0&0&0&0&5\\ \end{array} \right]$ ,

$\left[\begin{array}{ccccc} 2&1&0&0&0\\ 0&2&0&0&0\\ 0&0&2&0&0\\ 0&0&0&5&1\\ 0&0&0&0&5\\ \end{array} \right]$ , $\left[\begin{array}{ccccc} 2&1&0&0&0\\ 0&2&1&0&0\\ 0&0&2&0&0\\ 0&0&0&5&0\\ 0&0&0&0&5\\ \end{array} \right]$ , $\left[\begin{array}{ccccc} 2&1&0&0&0\\ 0&2&1&0&0\\ 0&0&2&0&0\\ 0&0&0&5&1\\ 0&0&0&0&5\\ \end{array} \right]$.