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 $x^{2}(x-1)^{2}$

2. Originally Posted by ypatia

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

3. Originally Posted by NonCommAlg
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??

4. Originally Posted by ypatia

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