1.8.7 Quadratic forms
condition: The quadratic form \( \varphi\left(x_{1}, x_{2}, x_{3}\right) \) is given. 1) Bring it to canonical form using the Lagrange method, writing out the corresponding transformation of variables. 2) Bring it to canonical form by an orthogonal transformation. 3) Check the law of inertia of quadratic forms using examples of transformations obtained in previous paragraphs 1)-2). 4) What surface is given by the equation \( \varphi\left(x_{1}, x_{2}, x_{3}\right)=1 \) ? \( \varphi\left(x_{1}, x_{2}, x_{3}\right)=2 x_{1}^{2}+9 x_{2}^{2}+2 x_{3}^{2}-4 x_{1} x_{2}+4 x_{2} x_{3} \).
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.