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Thread: find all possible Jordan Normal Forms of A

  1. #1
    Super Member bigwave's Avatar
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    find all possible Jordan Normal Forms of A

    This might be in the wrong forum
    But it is from a DE class

    Suppose that A is a matrix whose characteristic polynomial is
    $$(\lambda-2)^2(\lambda+1)^2$$
    find all possible Jordan Normal Forms of A (up to permutation of the Jordan blocks).


    ok i have been looking at examples so pretty fuzzy on this
    for the roots are 2 and -1


    so my first stab at this is

    $\left[\begin{array}{c} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 &2 \end{array}\right]$
    Last edited by bigwave; Mar 19th 2019 at 05:28 PM.
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  2. #2
    MHF Contributor

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    Re: find all possible Jordan Normal Forms of A

    Yes, that is one possible Jordan form. The general Jordan form has blocks for distinct eigenvalues with either 1 or 0 above each eigenvalue in each block. The other possible Jordan forms are
    $\displaystyle \begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 &0 & 2\end{bmatrix}$

    $\displaystyle \begin{bmatrix}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2\end{bmatrix}$

    $\displaystyle \begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2\end{bmatrix}$
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