How can I determine the Jordan Form of a matrix? How can I go about proving that the characteristic polynomial, minimal polynomial, and the dim(eigenspace) is enough to determine the Jordan Form of a matrix for n<7?

Let's give it a try:
$$A=\begin{pmatrix}\color{red}0&\color{red}1&\color{red}0&0&0&0&0\\
\color{red}0&\color{red}0&\color{red}1&0&0&0&0\\
\color{red}0&\color{red}0&\color{red}0&0&0&0&0\\
0&0&0&\color{red}0&\color{red}1&0&0\\
0&0&0&\color{red}0&\color{red}0&0&0\\
0&0&0&0&0&1&0\\
0&0&0&0&0&0&1\end{pmatrix}\;,\;\;\;
B=\begin{pmatrix}\color{blue}0&\color{blue}1&\color{blue}0&\color{blue}0&0&0&0&\\
\color{blue}0&\color{blue}0&\color{blue}1&\color{blue}0&0&0&0\\
\color{blue}0&\color{blue}0&\color{blue}0&\color{blue}1&0&0&0\\
\color{blue}0&\color{blue}0&\color{blue}0&\color{blue}0&0&0&0\\
0&0&0&0&\color{blue}0&0&0\\
0&0&0&0&0&1&0\\
0&0&0&0&0&0&1\end{pmatrix}$$
Clearly $\,A\nsim B\,$ since one has Jordan zero blocks of size $\,3,2\,$ resp. (in red) and the other one two Jordan zero blocks of size $\,4,1\,$ resp. (in blue) .
Both matrices have $\,x^5(x-1)^2\,$ as characteristic polynomial, both have $\,x^4(x-1)\,$ as minimal polynomial and both have zero eigenspace of dimension $\,5\,$ (check this!)
