1.8.14 Quadratic forms
condition: The quadratic form in some basis \( \left\{e_{1} ; e_{2} ; e_{3}\right\} \) has the form \( Q(x)=2 x_{1}^{2}+2 x_{1} x_{2}- \) \( -4 x_{2}^{2}-16 x_{1} x_{3} \). Find its matrix in the basis \( \left\{f_{1}=4 e_{1}-e_{3} ; f_{2}=e_{1}-2 e_{2} ; f_{3}=e_{2}-8 e_{3}\right\} \).
Investigation of quadratic forms, their reduction to canonical and normal forms with finding transformation matrices using Lagrange and orthogonal transformation methods. Positive and negative definite quadratic forms, Sylvester's criterion.