Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Classifying Symmetric Quadratic Forms
  1. Quadratic forms (Replies: 2)

  2. Quadratic forms (Replies: 1)

  3. Quadratic Forms (Replies: 5)

Loading...