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Orthonormal Basis

  1. Jan 13, 2008 #1
    1. The problem statement, all variables and given/known data
    My problem is I am getting a different answer than what MATLAB is giving me and I cannot determine why. Plz advise.

    Find an orthonormal basis of eigenvectors for matrix A= [3 2; 2 1] (using MATLAB notation- I couldn't figure out how to put in proper matrix notation).


    2. Relevant equations

    I find the eigenvectors as 4.2361 and -0.2361

    For eigenvalue 4.2361: [3 2; 2 1] - [4.2361 0; 0 4.2361] = [-1.2361 2; 2 -3.2361]
    -1.2361x1 + 2x2 = 0
    x2 = 0.6180 x1
    Therefore eigenvector: (1 0.6180) and normalizing it: (0.8506 0.5257)

    For eigenvector -0.2361: Eigenvector: (1 -1.6180) and normalizing it: (0.5257 -0.8506)

    Therefore my orthonormal basis of eigenvectors:
    (0.8506 0.5257; 0.5257 -0.8506)

    First Question:
    Is what the question is asking - to get an orthonormal basis of eigenvectors. Is this what I am doing?

    Second Question:
    I think it is but when I compare my answer to MATLAB, for eigenvector 4.2361, MATLAB gives normalized eigenvectors (-0.8506 -0.5257).
    I don't understand where the negative comes from.

    Thanks

    Asif
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 13, 2008 #2

    Tom Mattson

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    Those are the eigenvalues.

    You are obtaining an orthonormal basis of the eigenspace of the matrix.

    That's OK, the basis of the eigenspace is not unique. Think back to [itex]\mathbb{R}^2[/itex]. The standard basis is [itex]\{<1,0>,<0,1>\}[/itex], but an equally acceptable orthonormal basis is [itex]\{<-1,0>,<0,1>\}[/itex]. Both bases span the space, and they are both orthonormal.
     
  4. Jan 14, 2008 #3

    HallsofIvy

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    Is this a course on Linear Algebra/Matrix Algebra or a course on how to use MATLAB?

    The eigenvalues of this matrix are [itex]2\pm \sqrt{5}[/itex]. Do you know how to find those without using MATLAB?

    You seem to be consistently confusing "eigenvalues" with "eigenvectors"- you keep calling a single number an "eigenvector".

    An eigenvector corresponding to eigenvalue [itex]2+ \sqrt{5}[/itex] is [itex]<2, -1+\sqrt{5}>[/itex] and an eigenvector corresponding eigenvalue to [itex]2-\sqrt{5}[/itex] is [itex]<2, -1-\sqrt{5}>[/itex]. Can you find an orthonormal pair from that?
     
  5. Jan 14, 2008 #4
    HallsofIvy:

    - The course is on Linear Algebra. I use matlab to verify my results
    - I know how to compute eigenvectors/eigenvalues. It was a typing error when I called an eigenvalue an eigenvector.
    - As I said in original problem, I got the eigenvalues (same as you) and calculated the respective eigenvectors. I then found an orthonormal pair from it by squaring, summing and taking square root.

    - The thing I am not sure from T Mattson reply is that he is saying I am getting an orthonormal basis of eigenspace and not of eigenvector and I don't see the difference.

    If you can clarify I would appreciate.



    Thanks

    Asif
     
  6. Jan 14, 2008 #5

    Tom Mattson

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    The expression "orthonormal basis of eigenvectors" makes no sense. Spaces, not vectors, have bases.
     
  7. Jan 14, 2008 #6
    Tom-
    I will ask my professor but that is what he is asking in the h/w assignment.
    I quote: "Find an orthonormal basis of eigenvectors" for matrix which I gave above.

    Thanks

    Asif
     
  8. Jan 14, 2008 #7

    Tom Mattson

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    I'm pretty sure he meant "eigenspace", which is the space of all the eigenvectors of the matrix.
     
  9. Jan 14, 2008 #8

    HallsofIvy

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    An othonormal basis for the vector space consisting of eigenvectors of this matrix is what you really want to say.

    In this case, the 'eigenspace" is the entire space.
     
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