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Math Help - Jordan Forms of 6x6 matrix

  1. #1
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    Jordan Forms of 6x6 matrix

    How can I find all possible Jordan forms for a 6x6 matrix A having as minimal polynomial the x^{2}(x-1)^{2}
    Thanks anyone in advance.
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  2. #2
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    Quote Originally Posted by ypatia View Post

    How can I find all possible Jordan forms for a 6x6 matrix A having as minimal polynomial the x^{2}(x-1)^{2}
    Thanks anyone in advance.
    the largest 0-block and 1-block appearing in a Jordan normal form of A are B_0=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix} and B_1=\begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix} respectively. thus all possible forms are:

    \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: \begin{pmatrix}B_0 & & \\ & B_1 & \\ & & I \end{pmatrix}.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    the largest 0-block and 1-block appearing in a Jordan normal form of A are B_0=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix} and B_1=\begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix} respectively. thus all possible forms are:

    \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: \begin{pmatrix}B_0 & & \\ & B_1 & \\ & & I \end{pmatrix}.
    Thank you very much for your help but I'd like to ask you What is the e_{22} in the 4th matrix??
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  4. #4
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    Quote Originally Posted by ypatia View Post

    Thank you very much for your help but I'd like to ask you What is the e_{22} in the 4th matrix??
    e_{22}=\begin{pmatrix}0 & 0 \\ 0 & 1 \end{pmatrix}.
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