3
$\begingroup$

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?

  • 0
    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.2011-10-11

0 Answers 0