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NonCommAlg the largest 0-block and 1-block appearing in a Jordan normal form of $\displaystyle A$ are $\displaystyle B_0=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$ and $\displaystyle B_1=\begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}$ respectively. thus all possible forms are:
$\displaystyle \begin{pmatrix}B_0 & & \\ & B_0 & \\ & & B_1 \end{pmatrix}, \ \begin{pmatrix}B_0 & & \\ & B_1 & \\ & & B_1 \end{pmatrix}, \ \begin{pmatrix}B_0 & & \\ & B_1 & \\ & & 0 \end{pmatrix}, \ \begin{pmatrix}B_0 & & \\ & B_1 & \\ & & e_{22} \end{pmatrix},$ and: $\displaystyle \begin{pmatrix}B_0 & & \\ & B_1 & \\ & & I \end{pmatrix}.$