1.8.10 Quadratic forms

condition: Given a quadratic form \( Q(\vec{x}) \). 1) Bring \( Q(\vec{x}) \) to canonical form using the Lagrange method. Write down the appropriate transformation of variables. 2) Bring \( Q(\vec{x}) \) to canonical form using an orthogonal transformation, write out the transition matrix. 3) Verify the validity of the law of inertia of quadratic forms using the example of transformations obtained in paragraphs 1 and 2. 4) The second-order surface \( \sigma \) is defined in a rectangular Cartesian coordinate system by the equation \( Q(\vec{x})=\alpha \). Determine the type of surface \( \sigma \) and write its canonical equation. \[ Q(\vec{x})=x_{1}^{2}+x_{2}^{2}+2 x_{3}^{2}+4 x_{1} x_{2}+2 \sqrt{2} x_{1} x_{3}-2 \sqrt{2} x_{2} x_{3}, \quad \alpha=0 . \]