All possible Jordan normal forms with a given minimal polynomial 
Suppose $A \in M_{n\times n}(\Bbb R)$ has minimal polynomial $$(\lambda-1)^2(\lambda+1)^2$$ and $n\leq 6$. Find all possible Jordan normal forms.

If we denote the $n \times n$ Jordan block 
$$J_\alpha (n) = \begin{bmatrix}\alpha&1&\cdots&\cdots\\0&\alpha&1&\cdots\\\vdots&\vdots\\0&0&\cdots&\alpha\end{bmatrix}$$
Then for the $n=6$ case: the characteristic polynomial is 
$$(\lambda-1)^3(\lambda+1)^3$$ 
or 
$$(\lambda-1)^4(\lambda+1)^2$$
or 
$$(\lambda-1)^2(\lambda+1)^4$$
So are the possible Jordan normal forms the following? 
$$\left\{ \begin{bmatrix}J_1(3)&O\\O&J_{-1}(3)\end{bmatrix}, \begin{bmatrix}J_{-1}(3)&O\\O&J_{1}(3)\end{bmatrix}, \begin{bmatrix}J_1(4)&O\\O&J_{-1}(2)\end{bmatrix}, \\ \begin{bmatrix}J_{-1}(2)&O\\O&J_{1}(4)\end{bmatrix}, \begin{bmatrix}J_1(2)&O\\O&J_{-1}(4)\end{bmatrix}, \begin{bmatrix}J_{-1}(4)&O\\O&J_{1}(2)\end{bmatrix} \right\}$$

Any of the Jordan blocks cannot be of size $3$ (or more) because it would then correspond to a factor $(\lambda\pm 1)^{3\text{ or more}}$ in the minimal polynomial. However, to get to the minimal polynomial with factors $(\lambda\pm 1)^2$, there must be at least one Jordan block of size 2. Therefore, the solutions for $n=6$ are:
$$\left[\begin{array}{cc|cc|cc}1&1&0&0&0&0\\0&1&0&0&0&0\\\hline0&0&1&1&0&0\\0&0&0&1&0&0\\\hline0&0&0&0&-1&1\\0&0&0&0&0&-1\end{array}\right]$$
$$\left[\begin{array}{cc|c|c|cc}1&1&0&0&0&0\\0&1&0&0&0&0\\\hline0&0&1&0&0&0\\\hline0&0&0&1&0&0\\\hline0&0&0&0&-1&1\\0&0&0&0&0&-1\end{array}\right]$$
$$\left[\begin{array}{cc|cc|cc}1&1&0&0&0&0\\0&1&0&0&0&0\\\hline0&0&-1&1&0&0\\0&0&0&-1&0&0\\\hline0&0&0&0&-1&1\\0&0&0&0&0&-1\end{array}\right]$$
$$\left[\begin{array}{cc|cc|c|c}1&1&0&0&0&0\\0&1&0&0&0&0\\\hline0&0&-1&1&0&0\\0&0&0&-1&0&0\\\hline0&0&0&0&-1&0\\\hline0&0&0&0&0&-1\end{array}\right]$$
$$\left[\begin{array}{cc|c|cc|c}1&1&0&0&0&0\\0&1&0&0&0&0\\\hline0&0&1&0&0&0\\\hline0&0&0&-1&1&0\\0&0&0&0&-1&0\\\hline0&0&0&0&0&-1\end{array}\right]$$
