I construct a linear operator $T \in \mathcal{L(\mathbf{C}^7)}$ , where the minimal polynomial is $m_T (x) = x^2(x-1)^2$ and the caraterisitic polynomial is $p_T (x) = x^3(x - 1)^4$.
The linear operator that I construct is $$T: \begin{bmatrix} x_1\\x_2\\x_3\\x_4\\x_5\\x_6\\x_7\\ \end{bmatrix}\mapsto \begin{bmatrix} 0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&1&0&0&0\\ 0&0&0&1&1&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1 \end{bmatrix} \begin{bmatrix} x_1\\x_2\\x_3\\x_4\\x_5\\x_6\\x_7\\ \end{bmatrix}$$ I want to find an explicit direct sum decomposition of the space into T-cyclic subspaces