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Eigenfunctions and their Eigenvalues

  1. Oct 9, 2006 #1
    If I have two eigenfunctions of an operator with the same eigenvalue how do I construct linear combinations of my eigenfunctions so that they are orhtogonal?

    My eigenfunctions are: f=e^(x) and g=e^(-x)

    and the operator is (d)^2/(dx)^2
     
  2. jcsd
  3. Oct 9, 2006 #2

    dextercioby

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    Orthogonal wrt what?

    You need a scalar product.

    Daniel.
     
  4. Oct 9, 2006 #3
    I want to have linearly independent combinations of f and g that are orthognal on the interval from (-1,1) I'm guesing that they need to be wrt f and g.
     
  5. Oct 9, 2006 #4

    HallsofIvy

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    No, that was not the question. "Orthogonal" means that the inner product is 0 so whether or not two vectors are orthogonal depends on the inner product used.

    The most common inner product for real valued functions on an interval (a, b) is [itex]\int_a^b f(x)g(x)dx[/itex].

    Since, if two eigenvectors correspond to the same eigenvalue, any linear combination is also an eigenvector corresponding to that eigenvalue, a simple "orthogonal projection" will work.

    If u and v are two vectors in an inner product space, then the "projection of v onto u" is given by
    [tex]\frac{<u,v>}{<u,u>}\vec{u}[/tex]
    The "orthogonal projection" is v minus that:
    [tex]\vec{v}- \frac{<u,v>}{<u,u>}\vec{u}[/tex]

    Calculate that with u= ex, v= e-x, and inner product [itex]<u,v>= \int_{-1}^1 u(x)v(x)dx[/itex].
     
  6. Oct 9, 2006 #5
    O.k.

    I think that worked. I had been trying the integral in a slightly different way using f + g instead of fg.

    Thanks.
     
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