I’m trying to derive Sylvester’s criteria without using matrices in derivation, i.e.for two-dimensional space:
by definition of dot product and using Gram’s matrix,
$$(a,a)>0 \iff g_{11}a_1^2+2g_{12}a_1a_2+g_{22}a_2^2$$$$=g_{11}(a_1^2+\frac{2g_{12}}{g_{11}}a_1a_2+\frac{g_{22}}{g_{11}}a_2^2)= g_{11}(a_1+\frac{g_{12}}{g_{11}}a_2)^2+a_2^2(g_{22}-\frac{g_{12}^2}{g_{11}})$$
from where I get, that $g_{11}>0$ and $\det G>0$.
I tried to use the same methods for the three-dimensional space and factor my equation, but with no success. It looks like $(a,a)=g_{11}a_1^2+g_{22}a_2^2+g_{33}a_3^2+2g_{12}a_1a_2+2g_{13}a_1a_3+2g_{22}a_2a_3$.
How should I factor it?