# Thread: Jordan Forms of 6x6 matrix

1. ## Jordan Forms of 6x6 matrix

How can I find all possible Jordan forms for a 6x6 matrix A having as minimal polynomial the $\displaystyle x^{2}(x-1)^{2}$

2. Originally Posted by ypatia

How can I find all possible Jordan forms for a 6x6 matrix A having as minimal polynomial the $\displaystyle x^{2}(x-1)^{2}$
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: $\displaystyle \begin{pmatrix}B_0 & & \\ & B_1 & \\ & & I \end{pmatrix}.$

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

4. Originally Posted by ypatia

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