# P matrix of Jordan Form

• Dec 5th 2011, 05:35 AM
dwsmith
P matrix of Jordan Form
Find P such that, $\displaystyle P^{-1}AP=J$

$\displaystyle A=\begin{bmatrix}0&1&1&0&1\\0&0&1&1&1\\0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&0&0\end{bmatrix}$

$\displaystyle \lambda^5=0$

Minimal Polynomial is $\displaystyle \lambda^3-0$.

The Jordan Blocks are:

$\displaystyle \left\{\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmat rix} , \ \begin{bmatrix}0\end{bmatrix}, \ \begin{bmatrix}0\end{bmatrix}\right\}$

Eigenvectors:

$\displaystyle \left\{\begin{bmatrix}1\\0\\0\\0\\0\end{bmatrix}, \ \begin{bmatrix}0\\0\\-1\\0\\1\end{bmatrix}, \ \begin{bmatrix}0\\1\\-1\\1\\0\end{bmatrix}\right\}$

How do I find the other 2 vectors to complete my P matrix?

I have no idea why the LaTex is screwy on the first Jordan Block matrix but it is typed in correctly.
• Dec 5th 2011, 07:40 AM
FernandoRevilla
Re: P matrix of Jordan Form
Quote:

Originally Posted by dwsmith
How do I find the other 2 vectors to complete my P matrix?

If $\displaystyle u_1,u_2,u_3$ are respectively the eigen vectors you have written, find vectors $\displaystyle u'_1,u'_2$ such that $\displaystyle Au'_1=u_1$ and $\displaystyle Au'_2=u'_1$ . Then, $\displaystyle B_J=\{u_1,u'_1,u'_2,u_2,u_3\}$ is a basis of Jordan (why?).