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Linear algebra: orthonormal basis

  1. May 5, 2013 #1
    1. The problem statement, all variables and given/known data
    ##\phi## is an endomorphism in ##\mathbb{E}^3## associated to the matrix
    (1 0 0)
    (0 2 0) =##M_{\phi}^{B,B}##=
    (0 0 3)

    where B is the basis: B=((1,1,0),(1,-1,0),(0,0,-1))

    Find an orthonormal basis "C" in ##\mathbb{E}^3## formed by eigenvectors of ##\phi##

    3. The attempt at a solution

    Being the eigenvalues the elements of the diagonal 1, 2, 3
    Aren't (1, 0, 0), (0,2,0), (0,0,3) three orthonormal vectors already?

    Or should I write the endomorphism according to the canonical basis first and find new values?
     
  2. jcsd
  3. May 5, 2013 #2

    Mark44

    Staff: Mentor

    Yes, they are, but these aren't the vectors they're asking for.
    Use the eigenvalues to find a basis of eigenvectors, and then make an orthonormal basis out of that set of vectors.
     
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