# Quadratic transformations of vector spaces

Much is known about transformations of the following form $$y_i = L_{ij}x_j \;\;: \;\; x\in\mathcal{R}^n, L\in\mathcal{R}^{n\times n}$$

We can infer a number of geometric properties about the transformation from properties of the matrix $L_{ij}$ such as $$det(L)\ne 0 \implies \textrm{the transformation is invertible}$$ $$L \textrm{ orthogonal} \implies \textrm{angles and distances preserved}$$ and so forth. What do we know about transformations of the form $$y_i = Q_{ijk}x_jx_k$$

i.e. transformations which are quadratic in $x$. Is there a similar body of knowledge for these transformations?

If so by what name does it go by and what are good references/search terms?

If there is no such body of knowledge then why not?

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For example, every quadratic form Q induces a symmetric bilinear form B(x,y) = Q(x + y) - Q(x) - Q(y); about symmetric bilinear forms indeed much is known... – ItsNotObvious Oct 10 '11 at 19:34
Yes. Quadratic forms seem to be the closest thing that I've come across that is well studied. I would expect the above geometric problem to be common as well and am surprised as to why I'm having difficulty locating anything on it. – MRocklin Oct 11 '11 at 13:51