1.8.10 Quadratic Forms

Condition: given a quadratic form \ (q (\ vec {x}) \). 1) bring \ (q (\ vec {x}) \) to the canonical species by Lagrange. Record the appropriate transformation of the variables. 2) bring \ (q (\ vec {x}) \) to canonical appearance using an orthogonal transformation, write out the transition matrix. 3) to verify the justice of the law of inertia of quadratic forms on the example of transformations obtained in paragraphs 1 and 2. 4) the surface of the second order \ (\ sigma \) is specified in the rectangular cartmic system of the coordinates 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_} x_ {2} +2 \ sqrt {2} x_ {1} x_} -2 \ sqrt {2} x_ {2} x_ {3}, \ quad \ alpha = 0. \]