I need your help for understanding how to compute jordan base for this matrix:
$\begin{pmatrix} 2 &1 &2 \\ -1 &0 &2 \\ 0&0 & 1 \end{pmatrix}$
This is what I tried to do:
I found the Minimal polynomial: $(x-1)^{3}$,
so I know that the Jordan normal form is: $\begin{pmatrix} 1 &0 &0 \\ 1& 1 & 0\\ 0&1 & 1 \end{pmatrix}$
Now I remember that I need to find: $\ker (A-I), \ker (A-I)^{2}, \ker(A-I)^{3}$.
I found that $\ker (A-I)$ is: $\begin{pmatrix} -1\\ 1 \\ 0 \end{pmatrix}$ and $\ker (A-I)^{2}$ is: $\begin{pmatrix} 1\\ 0 \\ 0 \end{pmatrix}$ , $\begin{pmatrix} 0\\ 1 \\ 0 \end{pmatrix}$
and obivously $\ker(A-I)^{3}$ is all $\mathbb{R}^{3}$, What should I do from here? I can't remember the whole algorithem, and I can't find it anywhere.
Thank you, Have a great week-end.
You can choose a vector from $\ker(A-I)^{3}$ which is not in $\ker(A-I)^{2}$.
$\vec a=(0,0,1)^T$
$\vec b=(A-I)\vec a=A\vec a-\vec a=(2,2,0)^T$ belongs to $\ker(A-I)^{2}$ but not to $\ker(A-I)$.
$\vec c=(A-I)\vec b=(4,-4,0)$
The vectors $\vec c$, $\vec b$, $\vec a$ form a base in which the linear transformation $A$ has form $J$.
Two side notes:
The
minimal polynomialcharacteristic polynomial being $(x-1)^3$ does not determine the Jordan form uniquely. Only after you find out that the eigenspace has dimension 1 you know, that there is only one Jordan block. (EDIT: I meant the characteristic polynomial here - as pointed out in comments.)I have more often seen different form with 1's above the diagonal.