Find all possible Jordan canonical forms of $A$, where $(A-2I)^3(A+2I)^2=0$

Question

A $5\times 5$ matrix $A$ satisfies the equation $(A-2I)^3(A+2I)^2=0$. Assuming there are at least $2$ linearly independent eigenvectors for $2$, write all possible Jordan canonical forms.

My Question:

The eigenvalues of $A$ are $2$ and $-2$. If we know the number of linearly independent eigenvectors for $2$, can we determine the number of linearly independent eigenvectors for $-2$?

Answer 1

No: If we denote by $J_k(\lambda)$ the Jordan block of eigenvalue $\lambda$ and size $k \times k$, then both $$ J_1(2) \oplus J_1(2) \oplus J_1(2) \oplus J_1(-2) \oplus J_1(-2) \qquad\text{and} \qquad J_1(2) \oplus J_1(2) \oplus J_1(2) \oplus J_2(-2) $$ satisfy the equation $(A - 2I)^3 (A + 2I)^2 = 0$, and both have three linearly independent eigenvectors of eigenvalue $2$, but the first has two linearly independent eigenvectors of eigenvalue $-2$ and the second has only one.