1.8.15 Quadratic forms
condition: Reduce the quadratic form \( q\left(x_{1}, x_{2}, x_{3}\right) \) to canonical form by an orthogonal transformation (reduction to the principal axes). \[ \begin{array}{l} q\left(x_{1}, x_{2}, x_{3}\right)=x_{1}^{2}-5 x_{2}^{2}+x_{3}^{2}+4 x_{1} x_{2}+2 x_{1} x_{3}+ \\ +4 x_{2} x_{3} . \end{array}\]
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.