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Math Help - 4x4 Complex repeated Eigenvalues

  1. #1
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    4x4 Complex repeated Eigenvalues

    I'm trying to find the general solution for the following 4x4 matrix:
    [0 1 1 0]
    [-1 0 0 1]
    [0 0 0 1]
    [0 0 -1 0]

    So far, I have the eigenvalues as repeated i, i, - i ,-i
    Eigenvector for i: [-i 1 0 0]^t (with multiplicity 2)
    Eigenvector for -i: [1 i 0 0]^t (with multiplicity 2)

    How do I get the general solution for this? And how do I find the adjoint eigenvectors for a 4x4 with complex repeated eigenvalues?
    Last edited by Borkborkmath; February 27th 2011 at 03:30 PM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Borkborkmath View Post
    So far, I have the eigenvalues as repeated i, i, - i ,-i

    Right.


    How do I get the general solution for this?

    What do you mean by general solution?. General solution of ...
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  3. #3
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    Of the matrix,
    Umm.. I do believe it should look something like x(t) = c_1e^(eigen)(eigenvector) + ... + c_ne^(eigen)(eigenvector)

    This is from an ODE's class, but last time I posted a matrix question from ODE's a mod moved it and scolded me. So, that is why I put it in the linear algebra this time.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    I guess you mean the general solution of the differential system X'=AX. You can express


    X(t)=e^{tA}K\quad (K=(k_1,k_2,k_3,k_4)^t)

    or in a vectorial form:

    X(t)=k_1C_1(t)+k_2C_2(t)+k_3C_3(t)+k_4C_4(t)

    where C_i(t) are de columns of e^{tA} .
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  5. #5
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    Yes, thats exactly what I mean.

    How do I get the adjoint vectors of the repeated complex eigenvalues so that I can write the general solutions for X'= AX?
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Borkborkmath View Post
    Yes, thats exactly what I mean. How do I get the adjoint vectors of the repeated complex eigenvalues so that I can write the general solutions for X'= AX?

    Don't worry about it, all is included in the columns C_i(t) of e^{tA}.
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  7. #7
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    So the general solution is:
    X(t) = e^(tA)*(i 1 0 0) + e^(tA)*(1 i 0 0)?
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