Quadratic polynomial/R/Change of variables/Pure form/Fact/Proof

Proof

We apply the transformation described in fact to the pure-quadratic part ${}\sum _{i\leq j}a_{ij}X_{i}X_{j}$. In the new variables ${}V_{j}$ (which are dual to the orthonormal basis), the polynomial has now the form

${}F=\sum _{1\leq i\leq k}e_{i}V_{i}^{2}+\sum _{j=1}^{n}f_{j}V_{j}+g\,,$

with a certain ${}k$ between ${}1$ and ${}n$, and ${}e_{i}\neq 0$. The summands

$e_{i}V_{i}^{2}+f_{i}V_{i}$

can be brought, by completing the square and using new variables ${}U_{i}=V_{i}+h_{i}$, to the form

$e_{i}U_{i}^{2}+g_{i}.$

Besides the pure-quadratic term, either a constant or a linear polynomial remains. In the second case, we denote this linear form by ${}U_{k+1}$.

To fact