
1.8.1 Quadratic forms
Problem:
Investigate the quadratic form for signdefiniteness, bring it to the canonical form by orthogonal transformation and write down the transition matrix.
\[
\begin{array}{l}
f\left(x_{1}, x_{2} x_{3}\right)=3 x_{1}{ }^{2}+x_{2}{ }^{2}-\frac{3}{2} x_{3}{ }^{2}+2 \sqrt{3} x_{1} x_{2}- \\
-x_{1} x_{3}+\sqrt{3} x_{2} x_{3} .
\end{array}
\]