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Classifying Symmetric Quadratic Forms

  1. Aug 29, 2011 #1
    Hi, All:

    I am trying to see how to classify all symmetric bilinear forms B on R^3 as a V.Space
    over the reals.

    My idea is to use the standard basis for R^3 , then use the matrix representation M
    =x^T.M.y . Then, since M is, by assumption, symmetric, we can diagonalize M.

    So it seems all symmetric bilinear forms are just all diagonal matrices; maybe we
    need to factor out those that are equivalent as bilinear forms, i.e., B,B' are equivalent
    if there is a linear isomorphism L:R^3-->R^3 with B(v)=B'(L(v)).

    Is this right? Is there anything else?
  2. jcsd
  3. Aug 29, 2011 #2
    You can even do better!! You can even diagonalize those forms such that the diagonal matrix only has 1, -1 or 0 as diagonal entries!!
    A symmetric bilinear form with 0 as diagonal entry are exactly the non-degenerate forms.

    All this actually follows from Gram-Schmidt orthogonalization.

    See http://mathworld.wolfram.com/SymmetricBilinearForm.html
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