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Homework Help: Eigenvalue/Eigenvector proof

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    let p(x) = summation(from i=0 to k) aix^i

    matrix polynomial for A is defined as p(A) = summation(i=0 up to k) aiA^i

    Show that if (lambda, x) is an eigenpair of A then (p(lambda), x) is an eigenpair of p(A)

    2. Relevant equations



    3. The attempt at a solution

    I pretty much have no idea where to start. I thought I could use Ax = lambda x like you would if you were proving that lambda^2 is an eigenvalue of A^2, etc, but I'm not sure how to get the p(A) bit?
     
  2. jcsd
  3. Apr 27, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    try multiplying the polynomial matrix form by the eignvector & see the results
     
  4. Apr 27, 2010 #3

    lanedance

    User Avatar
    Homework Helper

    here's some latex to help, click on it to see code

    [tex] \sum_{i=0}^k a_i x^i [/tex]
     
  5. Apr 28, 2010 #4
    Thanks, I'm still not sure if I'm going about this the right way

    so i've got

    Use Ax = [tex]\lambda[/tex]x

    for p(A), multiply both sides of matrix polynomial by eigenvector x

    p(A)x =
    [tex]
    \sum_{i=0}^k a_i A^i
    [/tex] x

    from the polynomial in the question, p(x) =
    [tex]
    \sum_{i=0}^k a_i x^i
    [/tex]

    let x = [tex]\lambda[/tex]

    p([tex]\lambda[/tex]) =
    [tex]
    \sum_{i=0}^k a_i \lambda^i
    [/tex]

    p(A)x = [tex]
    \sum_{i=0}^k a_i \lambda^i
    [/tex] x

    p(A)x = p([tex]\lambda[/tex]x as required

    but i'm not sure if i've really shown anything. how do i show that its the same x and [tex]\lambda[/tex] that belong to the matrix A?
     
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