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Math Help - P matrix of Jordan Form

  1. #1
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    P matrix of Jordan Form

    Find P such that, P^{-1}AP=J

    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}

    \lambda^5=0

    Minimal Polynomial is \lambda^3-0.

    The Jordan Blocks are:

    \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:

    \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.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: P matrix of Jordan Form

    Quote Originally Posted by dwsmith View Post
    How do I find the other 2 vectors to complete my P matrix?
    If u_1,u_2,u_3 are respectively the eigen vectors you have written, find vectors u'_1,u'_2 such that Au'_1=u_1 and Au'_2=u'_1 . Then, B_J=\{u_1,u'_1,u'_2,u_2,u_3\} is a basis of Jordan (why?).
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