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Finding eigenlines & eigenvalues

  1. Feb 3, 2012 #1
    In my example I have matrix A = (1 2)
    . . . . . . . . . . . . . . . . . . . . . . (3 2)

    Finding the eigenvalue through the method I understand & can get the result

    i.e.
    k = 4 & -1

    I suspect my algebra is the shaky link, here, but to find the eigenline I find a bit more of a challenge.

    OK I start by substituting the eigenvalue into the eigenvector equation;
    Ax = kx
    giving
    (1 2) (x) = 4 (x)
    (3 2) (y)......(y)


    which gives rise to the following simultaneous equations
    x + 2y = 4x
    3x + 2y = 4y

    Now the bit I don't get . . .
    How do these both reduce to 3x - 2y = 0

    I'm sure it's simple & I can't see the wood for the trees, but this is stupidly defeating me.. . . . Grrr!

    Please could someone point me in the right direction.
     
    Last edited: Feb 3, 2012
  2. jcsd
  3. Feb 3, 2012 #2
    Damn, it's so easy to suddenly see the solution.
    OK I'm done

    x + 2y = 4x
    3x+ 2y =4y

    goes to
    -3x + 2y = 0
    3x - 2y = 0

    both reduce to
    3x - 2y = 0

    Thank you.
     
  4. Feb 5, 2012 #3
    All the other examples I have had led to the simultaneous equations reducing to a single equation, from which I can find the eigenline equation for each eigenvalue (2 in this case).

    The simultaneous equation I have now is;
    3/4 x - 3/4 y = 0
    -1/4 x - 1/4 y = 0

    I am assuming I have the negative value wrong, but I may not, I suppose, so how can I get back to a single equation so I can find the eigenline equation for each eigenvalue.
     
  5. Feb 5, 2012 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Please tell us what the matrix in this problem is so that we can tell whether or not 2 really is an eigenvalue.
     
  6. Feb 6, 2012 #5
    (11/4 -3/4)
    (-1/4 9/4)

    This is the original matrix. Thank you.
     
  7. Feb 6, 2012 #6

    Deveno

    User Avatar
    Science Advisor

    based on the matrix you provided, the second equation should be:

    -(1/4)x + (1/4)y = 0.

    thus BOTH equations lead to:

    x = y.
     
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