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Jordan Forms for a 3x3 Matrix

  1. Mar 12, 2012 #1
    1. The problem statement, all variables and given/known data
    1. The problem statement, all variables and given/known data[/b]
    Enumerate all possible Jordan forms for 3 x 3 systems where all the eigen-values have negative real parts. Do not use specific values. Instead, use possibilities
    like λ1; λ2; λ3, each with multiplicity 1, or λ (multiplicity 3).



    2. Relevant equations



    3. The attempt at a solution

    Let Ji be the Jordan Form

    J1=\begin{bmatrix}
    λ1 & 0 & 0 \\
    0 & λ2 & 0\\
    0 & 0 & λ3
    \end{bmatrix}

    So λ1, λ2, and λ3 all have multiplicity 1

    J2=\begin{bmatrix}
    λ1 & 0 & 0 \\
    0 & λ2 & 1\\
    0 & 0 & λ2
    \end{bmatrix}

    λ1 (Multiplicity 1), λ2 (Multiplicity 2)


    J3=\begin{bmatrix}
    λ1 & 0 & 0\\
    0 & λ1 & 0\\
    0 & 0 & λ1
    \end{bmatrix}

    λ1 (Multiplicity 3) With 1 generalized eigenvector



    J4=\begin{bmatrix}
    λ1 & 1 & 0\\
    0 & λ1 & 1\\
    0 & 0 & λ1
    \end{bmatrix}

    λ1 (Mulitiplicity 3) With 2 generalized eigenvectors


    J5=\begin{bmatrix}
    λ1 & 0 & 0 \\
    0 & λ2 & 0\\
    0 & 0 & λ3
    \end{bmatrix}

    Where λ1 ε ℝ, λ2 and λ3 are complex conjugates such that
    λ2= -a+bi and λ3=-a-bi. So λ1, λ2, and λ3 all have multiplicity 1.


    1) Do these Jordan Forms look correct?
    2) Are there more? ( I think there may be 3 more but I'm unsure)
     
  2. jcsd
  3. Mar 13, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Why the blank spaces? Were those supposed to be "1"s?
     
  4. Mar 13, 2012 #3
    Sorry if I seem confused but what blank spaces?
     
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