Classifying Symmetric Quadratic Forms

[Bacle]

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?

 

[micromass]

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