Homework 6
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(30 p.) Describe all possible Jordan forms for a nilpotent linear operator on a finite-dimensional vector space \(V\) with \(\dim V \leq 6\).
Since the matrix is nilpotent, then \(\chi_{f}(x)=x^{\dim V}\)
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\(\dim V=6\)
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One block: \(\begin{pmatrix} 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} \\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} \\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} \\ \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} \\ \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} \\ \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0 \end{pmatrix}\)
- Two blocks: $ \begin{pmatrix}\begin{array}{ccccc|c} 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} \ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} \ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} \ \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} \ \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} \\hline \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}$ or \(\begin{pmatrix} \begin{array}{cccc|cc}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & & 0 & \placeholder{}\\ \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0\end{array} \end{pmatrix}\) or \(\begin{pmatrix} \begin{array}{ccc|ccc}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & & 0 & \placeholder{} & \placeholder{}\\ \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{}\\ \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0\end{array} \end{pmatrix}\)
- Three blocks: \(\begin{pmatrix} \begin{array}{cccc|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{}\\\hline \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & 0 & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\) or \(\begin{pmatrix} \begin{array}{ccc|cc|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\\hline \placeholder{} & \placeholder{} & & 0 & \placeholder{} & \placeholder{}\\ \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\) or \(\begin{pmatrix} \begin{array}{cc|cc|cc}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & 0 & \placeholder{}\\ \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0\end{array} \end{pmatrix}\)
- Four blocks: \(\begin{pmatrix} \begin{array}{ccc|c|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & 0 & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\) or \(\begin{pmatrix} \begin{array}{cc|cc|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & & 0 & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\)
- Five blocks: \(\begin{pmatrix} \begin{array}{cc|c|c|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & & 0 & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\)
- Six blocks: \(\begin{pmatrix} \begin{array}{c|c|c|c|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\\hline & 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & & 0 & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\)
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\(\dim V=5\)
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One block:\(\begin{pmatrix} 0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} \\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{} \\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{} \\ \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} \\ \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0 \end{pmatrix}\)
- Two blocks: \(\begin{pmatrix} \begin{array}{cccc|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{} \\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & 1 & 0 & \placeholder{}\\ \placeholder{} & \placeholder{} & 1 & 0 & \placeholder{} \\ \hline\placeholder{} & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\) or \(\begin{pmatrix} \begin{array}{ccc|cc}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & & 0 & \placeholder{}\\ \placeholder{} & \placeholder{} & \placeholder{} & 1 & 0\end{array} \end{pmatrix}\)
- Three blocks: \(\begin{pmatrix} \begin{array}{ccc|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \placeholder{} & 1 & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & & 0 & \placeholder{}\\ \hline & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\) or \(\begin{pmatrix} \begin{array}{cc|cc|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline & & 0 & \placeholder{} & \placeholder{}\\ & \placeholder{} & 1 & 0 & \placeholder{}\\ \hline & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\)
- Four blocks: \(\begin{pmatrix} \begin{array}{cc|c|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline & & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & & 0 & \placeholder{}\\ \hline & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\)
- Five blocks: \(\begin{pmatrix} \begin{array}{c|c|c|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline & 0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline & & 0 & \placeholder{} & \placeholder{}\\ \hline\placeholder{} & \placeholder{} & & 0 & \placeholder{}\\ \hline & \placeholder{} & \placeholder{} & & 0\end{array} \end{pmatrix}\)
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\(\dim V=4\)
- One block: \(\begin{pmatrix} 0 & \placeholder{} & \placeholder{} & \placeholder{} \\ 1 & 0 & \placeholder{} & \placeholder{} \\ \placeholder{} & 1 & 0 & \placeholder{} \\ \placeholder{} & \placeholder{} & 1 & 0 \end{pmatrix}\)
- Two blocks: \(\begin{pmatrix} \begin{array}{ccc|c}0 & \placeholder{} & \placeholder{} & \placeholder{} \\ 1 & 0 & \placeholder{} & \placeholder{} \\ \placeholder{} & 1 & 0 & \placeholder{} \\ \hline & \placeholder{} & & 0\end{array} \end{pmatrix}\) or \(\begin{pmatrix} \begin{array}{cc|cc}0 & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{}\\ \hline & & 0 & \placeholder{}\\ & \placeholder{} & 1 & 0\end{array} \end{pmatrix}\)
- Three blocks: \(\begin{pmatrix} \begin{array}{cc|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{}\\ 1 & 0 & \placeholder{} & \placeholder{}\\ \hline & & 0 & \placeholder{}\\ \hline & \placeholder{} & & 0\end{array} \end{pmatrix}\)
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Four blocks: \(\begin{pmatrix} \begin{array}{c|c|c|c}0 & \placeholder{} & \placeholder{} & \placeholder{}\\ \hline & 0 & \placeholder{} & \placeholder{}\\ \hline & & 0 & \placeholder{}\\ \hline & \placeholder{} & & 0\end{array} \end{pmatrix}\)
- \(\dim V=3\)
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One block: \(\begin{pmatrix} 0 & \placeholder{} & \placeholder{} \\ 1 & 0 & \placeholder{} \\ \placeholder{} & 1 & 0 \end{pmatrix}\)
- Two blocks:\(\begin{pmatrix}\begin{array}{cc|c} 0 & \placeholder{} & \placeholder{} \\ 1 & 0 & \placeholder{} \\ \hline & & 0\end{array} \end{pmatrix}\)
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Three blocks: \(\begin{pmatrix} \begin{array}{c|c|c}0 & \placeholder{} & \placeholder{}\\ \hline & 0 & \placeholder{}\\ \hline & & 0\end{array} \end{pmatrix}\)
- \(\dim V=2\)
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One block: \(\begin{pmatrix} 0 & \placeholder{} \\ 1 & 0 \end{pmatrix}\)
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Two blocks: \(\begin{pmatrix}\begin{array}{c|c} 0 & \placeholder{} \\ \hline & 0\end{array} \end{pmatrix}\)
- \(\dim V=1\)
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One block: \((0)\)
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(35 p.) Find the nilpotent Jordan form and a corresponding Jordan basis for the linear operator \(f\) on \(K^8\) defined by the matrix \(\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & -1 & 0 & 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}\)
\(\chi_{f}(x)=\det \begin{pmatrix} x & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & x & 0 & -1 & -1 & 0 & 0 & 0 \\ 0 & 0 & x & 0 & 0 & 0 & -1 & 0 \\ -1 & 0 & 1 & x & 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 & x & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & x & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & x \end{pmatrix}=x^{8}\)
Thus this matrix is nilpotent. Then we find \(m_A(x)\) with standard basis
\(e_{1}\to-e_{2}+e_{4}-e_{5}\to e_{5}-e_{6}\to e_{2}+e_{6}\to0\)
\(e_2\to 0\)
\(e_{3}\to-e_{4}\to-e_{2}-e_{5}\to-e_{2}-e_{6}\to0\)
\(e_{4}\to e_{2}+e_{5}\to e_{2}+e_{6}\to0\)
\(e_{5}\to e_{2}+e_{6}\to0\)
\(e_6\to 0\)
\(e_{7}\to e_{3}+e_{6}+e_{8}\to0\)
\(e_{8}\to e_{4}\to e_{2}+e_{5}\to e_{2}+e_{6}\to0\)
Then \(m_{e_1}=x^{4},m_{e_2}=x,m_{e_3}=x^{4},m_{e_4}=x^{3},m_{e_5}=x^{2},m_{e_6}=x,m_{e_7} =x^{2},m_{e_8}=x^{4}\)
Then \(m_{A}(x)=x^{4}\) since the least common multiple
Thus the largest Jordan elementary block is \(4\times 4\)
And since \(\text{rank}A=4\), then \(\ker f=4\), thus \(\ker f=\{e_{2},e_{6},e_{3}+e_{8},e_{1}+e_{4}+e_{3}\}\)
Thus the only possibility is \(\dim\ker f=4,\dim\ker f^{2}=6,\dim\ker f^{3}=7,\dim\ker f^{4}=8\)
Then \(\ker f^{2}=\{e_{2},e_{6},e_{3}+e_{8},e_{1}+e_{4}+e_{3},e_{5},e_{7}\},\ker f^{3}= \{e_{2},e_{6},e_{3}+e_{8},e_{1}+e_{4}+e_{3},e_{5},e_{7},e_{4}\},\ker f^{4}=\{e_{2} ,e_{6},e_{3}+e_{8},e_{1}+e_{4}+e_{3},e_{5},e_{7},e_{4},e_{3}\}\)
And we have \(\{0\}\subsetneq\ker A\subsetneq\ker A^{2}\subsetneq\ker A^{3}\subsetneq\ker A^{4} =\mathbb{K}^{8}\)
\(0\leftarrow e_{2}+e_{6}\leftarrow e_{5}-e_{6}\leftarrow-e_{2}+e_{4}-e_{5}\leftarrow e_{1}\)
\(0\leftarrow e_{3}+e_{6}+e_{8}\leftarrow e_{7}\)
\(0\leftarrow e_{1}+e_{3}+e_{4}\)
\(0\leftarrow e_2\)
Thus the Jordan basis is \(B=\{e_{1},-e_{2}+e_{4}-e_{5},e_{5}-e_{6},e_{2}+e_{6},e_{7},e_{3}+e_{6}+e_{8},e_{1} +e_{3}+e_{4},e_{2}\}\)
Thus \(J= \begin{pmatrix} \begin{array}{cccc|cc|c|c}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array} \end{pmatrix}\)
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(35 p.) Find all nilpotent Jordan forms of \(A \in \mathbb{R}^{13 \times 13}\) given that \(\dim \ker A = 5\) and \(x^6\) divides the minimal polynomial \(m_A(x)\).
Since \(\dim\ker A=5\), then the number of Jordan blocks is \(5\)
We know \(x^6\) divides \(m_A(x)\), thus \(m_A(x)=x^i\) where \(i\geq 6\)
Then the biggest size of Jordan block is greater than \(6\)
Thus it can be 6 4 1 1 1, 6 3 2 1 1, 6 2 2 2 1 or 7 3 1 1 1, 7 2 2 1 1 or 8 2 1 1 1 or 9 1 1 1 1
Thus we have following Jordan forms of \(A\)
\(J_{1}= \begin{pmatrix} \begin{array}{cccccc|cccc|c|c|c}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array} \end{pmatrix}\)
\(J_{2}= \begin{pmatrix} \begin{array}{cccccc|ccc|cc|c|c}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array} \end{pmatrix}\)
\(J_{3}= \begin{pmatrix} \begin{array}{cccccc|cc|cc|cc|c}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array} \end{pmatrix}\)
\(J_{1}= \begin{pmatrix} \begin{array}{cccccc|cccc|c|c|c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \end{pmatrix}\)
\(J_{1}= \begin{pmatrix} \begin{array}{cccccc|cccc|c|c|c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \end{pmatrix}\)
\(J_{1}= \begin{pmatrix} \begin{array}{cccccc|cccc|c|c|c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \end{pmatrix}\)
\(J_{7}= \begin{pmatrix} \begin{array}{ccccccccc|c|c|c|c}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array} \end{pmatrix}\)