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?