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Homework Help: Problem about Eigenvalues

  1. Jul 17, 2012 #1
    1. The problem statement, all variables and given/known data


    A linear operator given (a matrix). That could be an orthogonal protection (that goes through the origin) or a symmetry with respect to a plane (that goes through the origin).

    1-Get the eigenvalues of linear operator
    2-Get the eigenspace associated with each eigenvalue.

    3-Based on the previous calculations determine if the Operator is a symmetry or an orthogonal protection.

    4-Describe an ortogonal base of the given plane, and complete it with a base of R^2
    The matrix with respect to the calculated base must have the form of the orthogonal projection or of the symmetric matrix

    100 or 100
    010 010
    00-1 000
    2. Relevant equations



    3. The attempt at a solution

    I got the eigenvalues 1 and 0 therefore I'm assuming the operator is an orthogonal projection.

    I got the eigenvectors

    How can I start to do 4?

    Im thinking about using gram schidt to get 3 ortogonal vectors and then to use them as a base .

    Thanks a lot for any help, I appreciate it.
     
  2. jcsd
  3. Jul 17, 2012 #2

    ehild

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    What is the given operator?


    ehild
     
  4. Jul 17, 2012 #3
    It is the matrix
    2 1 -1
    -1 0 1
    1 1 0
     
  5. Jul 17, 2012 #4

    HallsofIvy

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    Okay, so this is projection onto a line? What is that line? What is a vector in the direction of that line?
     
  6. Jul 17, 2012 #5
    It seems to be is an orthogonal projection on a plane that goes through the origin.

    For now Im confused, I was thinking about using the set of eigenbasis or eigenvectors of the linear operator B . and to use gram schmidt to get 3 orthogonal bases w1, w2 and w3

    Im considering to evaluate w1 w2 and w3 with respect to the basis B to get the projection of the plane with respect to B.
     
  7. Jul 17, 2012 #6

    HallsofIvy

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    Once again, what are the eigenvectors corresponding to each eigenvalue?
     
  8. Jul 17, 2012 #7
    the eigenvectors associated with the eigenvalue 1 are -1,1,0 and 1 0 1 .
    the ones associated with the eigenvalue 0 are 1 -1 1
     
  9. Jul 18, 2012 #8

    ehild

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    Any linear combination of the eigenvectors belonging to 1 is also an eigenvector to λ=1. Find a combination of a=(1,0,1) and b=(-1,1,0) c=a+kb so the dot product a˙c=0 and choose a and c as orthogonal base in the plane.

    ehild
     
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