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The eigenvalues and eigenvectors of T

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  1. Aug 2, 2016 #1
    1. The problem statement, all variables and given/known data
    The eigenvectors and eigenvalues of T.PNG

    2. Relevant equations
    The lattice laplacian is defined as [itex] \Delta^2 = \frac{T}{\tau} [/itex], where T is the transition matrix [tex]
    \left[ \begin{array}{cccc}
    -2 & 1 & 0 & 0 \\
    1 & -2 & 1 & 0 \\
    0 & 1 & -2 & 1 \\
    0 & 0 & 1 & -2 \end{array} \right]
    [/tex]

    and [tex]\tau[/tex] is a time constant, which is taken = 1.

    3. The attempt at a solution
    The eigenvectors and eigenvalues of T solution.PNG
     
  2. jcsd
  3. Aug 2, 2016 #2

    Ssnow

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    remember that the eigenvector is a vector ##v=[O_{1}(j),O_{2}(j),O_{3}(j),O_{4}(j)]## (for the case ##4\times 4##), ##\tau=1##, so is ##v\Delta^{2}=\lambda v## ...
     
  4. Aug 2, 2016 #3
    Right.

    How is the lattice Laplacian commonly defined?
     
    Last edited: Aug 2, 2016
  5. Aug 2, 2016 #4

    Ssnow

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    Gold Member

    Sincerely I searched on the web and I have found only this

    https://en.wikipedia.org/wiki/Discrete_Laplace_operator

    this is the discrete Laplace operator and yours...

    Regarding your problem (in the example that you proposed) is the same to verify the system:



    ##[O_{1}(j),O_{2}(j),O_{3}(j),O_{4}(j)]\left[\begin{array}{cccc}-2 & 1 & 0&0 \\ 1&-2&1&0\\0&1&-2&1\\0&0&1&-2 \end{array}\right]=[\lambda O_{1}(j),\lambda O_{2}(j),\lambda O_{3}(j),\lambda O_{4}(j)]##

    that is

    ##-2O_{1}+O_{2}=\lambda O_{1}, O_{1}-2O_{2}+O_{3}=\lambda O_{2}, ... ## and so on ...
     
  6. Aug 2, 2016 #5
    Okay, I knew that definition of the lattice Laplacian. It's what we used in the course, but it was not defined as such.

    Right.

    Next step is solving the characteristic equation for the eigenvalues of the system.
     
  7. Aug 2, 2016 #6

    Ssnow

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  8. Aug 2, 2016 #7
    I read it through, but we haven't worked that much in-depth with eigenvalues and eigenvectors with respect to the lattice Lalplacian or used applied linear algebra sufficiently for me to easily understand that. So it's slightly above my mathematical skills, but I'll talk it through with my professor tomorrow and see what I come up with.

    But from what I could garner, the eigenvalues are given simply by two formulas, one for the even-valued k and one for odd-valued k.
     
    Last edited: Aug 2, 2016
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