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Expressing a symmetric matrix in terms on eigenvalues/vectors

  1. May 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Generate a random 10 x 10 symmetric matrix A (already done in MATLAB) . Express A in the form


    2. Relevant equations

    ## A = ## [itex]\displaystyle \sum_{j=1}^{10}λ_j(A)v_jv^T_j\ [/itex]

    for some real vectors ##v_j, j = 1, 2, . . . , 10.##

    3. The attempt at a solution

    I'm pretty sure the solution has something to do with the eigenvectors of a symmetric matrix being orthogonal.

    The whole sum is basically ##A = λ_1v_1v^T_1 + λ_2v_2v^T_2 . . . λ_{10}v_{10}v^T_{10}.##

    If ##v_j## are an eigenvectors then we can express that as:

    ##A = Av_1v^T_1 + Av_2v^T_2 . . . Av_{10}v^T_{10} ##
    ##→ A = A( v_1v^T_1 + v_2v^T_2 . . . v_{10}v^T_{10} ) ##

    But that means we need ## v_1v^T_1 + v_2v^T_2 . . . v_{10}v^T_{10} = I ##. But I don't know if these vectors are meant to have those properties. =S

    Basically I need to know what the ##v_j ##'s are
     
  2. jcsd
  3. May 9, 2013 #2
    Ok, so after testing some symmetric 3x3 matrices and computing it by hand I can confirm that the ##v_j##'s are indeed the eigenvectors of ##A##.

    Now I just need to know why, lol.
     
  4. May 10, 2013 #3
    Does anyone know why?

    If A is a symmetric matrix then A can be expressed as:

    ## A = ## [itex]\displaystyle \sum_{j=1}^{n}λ_j(A)v_jv^T_j\ [/itex]

    where ##v_j, j = 1, 2, . . . , n.## are the eigenvectors of ##A##

    But why???
     
  5. May 10, 2013 #4

    Dick

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    Science Advisor
    Homework Helper

    Your ##v_j## are an orthonormal set of eigenvectors, right? Multiply ##sum_{j=1}^{n}λ_j(A)v_jv^T_j## by any eigenvector ##v_k##. You get the same thing as multiplying that vector by A, right? If you want a fancy name for it, it's a simple version of the spectral theorem.
     
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