1.8.i Quadratic forms
Condition: Reduce the quadratic form \( F(X)=X^{T} A X \) to the canonical form \( F(Y)=Y^{T} A Y \). Find the matrix \( C \) of the transformation \( X=C Y \), reducing \( F(X) \) to the form \( F(Y) \). \[ F(X)=x_{1}^{2}-2 x_{1} x_{2}-2 x_{1} x_{3}+3 x_{2}^{2}-2 x_{2} x_{3}+2 x_{3}^{2} \text {. } \]
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.