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Eigenvalues and Normalised Eigenvectors

  1. May 4, 2010 #1
    1. The problem statement, all variables and given/known data
    I have a matrix
    H= [h g
    g h]
    and I need to find the eigenvalues and normalised eigenvectors


    2. Relevant equations



    3. The attempt at a solution
    I subtracted lamda from the diagonal and then solved for the determinant equally zero. The eigenvalues I found were
    (h-lambda)^2=g^2
    so (h-lambda)=+/- g
    lamdba=h+/-g

    but I'm not sure how to find the normalised eigenvectors?
     
  2. jcsd
  3. May 4, 2010 #2

    radou

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    After finding the eigenvalues, plug them back into the equation (A - λ I)x = 0, one by one, to get your eigenvectors. Then normalize them.
     
  4. May 4, 2010 #3
    Thank you very much.

    So what I have done is
    for eigenvalue h+g I have two equations
    hx+gy=hx+gx and gx+hy = hy+gy which gives x=y so eigenvalue h+g has eigenvector (1,1)
    and for eigenvalue h-g I have two equations
    hx+gy=hx-gx and hx+gy=hy-gy so x=-y so eigenvalue h-g has eigenvector (1,-1)

    Is that correct?
     
  5. May 4, 2010 #4
    are these normalised?
     
  6. May 5, 2010 #5

    radou

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    What does it mean for a vector to be normalized?
     
  7. May 5, 2010 #6
    to be of unit length...which these are right?
     
  8. May 5, 2010 #7

    radou

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    I am not going to check your calculations, but with respect to the standard norm neither (1, 1) nor (1, -1) have unit length, since |(1, 1)|=(|(1, -1)|) = √2.
     
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