Alll the matrices $A\in M_{7x7}\left(\mathbb{C}\right)$, with characteristic polynomial is: $\left(x-1\right)^3\left(x-2\right)^4$, ...

Question

I need to find all the matrices $A\in M_{7x7}\left(\mathbb{C}\right)$, all I know is the characteristic polynomial is: $$\left(x-1\right)^3\left(x-2\right)^4$$ $$\dim\:\ker\:\left(A-2I\right)=3$$ $$\dim\:\ker\:\left(A-I\right)=2$$

So as we know from Jordan Normal Form the Jordan block size of eigenvalue 1 is 3 and the Jordan block size of eigenvalue $2$ is $4,$ so for $1$ it can be $(2,1),(1,1,1)$ and for $2$ it can be $(2,1,1), (3,1), (2,2).$ where the tuples are the possible Jordan blocks forms sizes.

Which information from this can I use to find the Jordan Normal Form ?

Answer 1

I get $\begin{pmatrix} 2&0&0&0&0&0&0\\0&2&1&0&0&0&0\\0&0&2&0&0&0&0\\0&0&0&2&0&0&0\\0&0&0&0&1&1&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\end{pmatrix}$, up to similarity. The geometric multiplicity of $2$ is $3$. That of $1$ is $2$.