<s>[INST] How many possible Jordan decompasitions for an Endomorphism? 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 $\phi$?
 [/INST] 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
$\left[\begin{array}{ccccc}
2&0&0&0&0\\
0&2&0&0&0\\
0&0&2&0&0\\
0&0&0&5&0\\
0&0&0&0&5\\
\end{array}
\right]$
,
$\left[\begin{array}{ccccc}
2&0&0&0&0\\
0&2&0&0&0\\
0&0&2&0&0\\
0&0&0&5&1\\
0&0&0&0&5\\
\end{array}
\right]$
,
$\left[\begin{array}{ccccc}
2&1&0&0&0\\
0&2&0&0&0\\
0&0&2&0&0\\
0&0&0&5&0\\
0&0&0&0&5\\
\end{array}
\right]$
,
$\left[\begin{array}{ccccc}
2&1&0&0&0\\
0&2&0&0&0\\
0&0&2&0&0\\
0&0&0&5&1\\
0&0&0&0&5\\
\end{array}
\right]$
,
$\left[\begin{array}{ccccc}
2&1&0&0&0\\
0&2&1&0&0\\
0&0&2&0&0\\
0&0&0&5&0\\
0&0&0&0&5\\
\end{array}
\right]$
,
$\left[\begin{array}{ccccc}
2&1&0&0&0\\
0&2&1&0&0\\
0&0&2&0&0\\
0&0&0&5&1\\
0&0&0&0&5\\
\end{array}
\right]$.
</s>