$A =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}$, $B =\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$,
$C =\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$, $D =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$,
$E =\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$, $F =\begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}$,
So far, I have found that $B$ is similar to $C$ and $A$ is similar to $E$ by using permutation matrices. One thing that is confusing me is that we're working in $\mathbb{R}$. I know that $A$ and $B$ are in Jordan Canonical Form and are therefore similar over $\mathbb{C}$. Can we say the same for $\mathbb{R}$? Also, I found that $F$ is not similar to any of the other matrices, since its minimal polynomial is degree 3 as opposed to degree 2. Any ideas on how to check if $D$ is similar to another matrix? Thanks in advance.
$$ \frac{1}{2} \left( \begin{array}{ccc} 2&0&0\\ 0&1&0\\ 0&0&2\\ \end{array} \right) \left( \begin{array}{ccc} 1&0&0\\ 0&1&2\\ 0&0&1\\ \end{array} \right) \left( \begin{array}{ccc} 1&0&0\\ 0&2&0\\ 0&0&1\\ \end{array} \right) = \left( \begin{array}{ccc} 1&0&0\\ 0&1&1\\ 0&0&1\\ \end{array} \right) $$