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Finding eigenvalues

  1. Oct 30, 2005 #1
    I have the tridiagonal matrix (which comes from the backward Euler scheme)

    A =
    [ 1+2M - M 0 .... ]
    [ -M 1+2M 0 .... ]
    [ ..... ]
    [ -M 1+2M ]


    I am given that the eigenvectors are v_k = [ sin(pi*k*x_j ] for j = 1:J-1 and k = 1:J-1

    I am to find the eigenvalues L_k.

    I tried just writing A * v_k = L_k * v_k

    Then doing the multiplication on the left hand side I can get a relationship from one equation:

    -M*sin(sin(pi*k*x_j-1) + (1+2M)sin(pi*k*x_j) -M*sin(pi*k*x_j+1) = L_k * sin(pi*k*x_j)

    I get stuck here. Is this the right approach? Is there someway to solve for L_k?


    Also: I am to show the v_k are orthogonal. Would I use dot product or the integral inner product to show this?

    Thanks for any help.
     
  2. jcsd
  3. Oct 31, 2005 #2

    CarlB

    User Avatar
    Science Advisor
    Homework Helper

    You've got the right answer. If you think the instructor wants you to reduce it to simplest form, then look up the trig identity for sin(A+B) to bring those nasty sines into more equal terms.

    I would guess that they're wanting you to use the integral inner product. That is, they want you to integrate over x the dot product. But the nature of dot produts should give you that the dot product itself is zero for all x so your integral will be trivial.

    Good luck. And if you want your questions answered faster, try learning LaTex and making your questions pretty.

    Carl
     
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