= Solution
Complete the square in the <quadratic form>:
$$
Q_1=(x_1+2x_2)^2+x_2^2+6x_3^2.
$$
The real invertible <linear transformation>
$$
\boxed{x=Ly,\qquad L=\begin{pmatrix}1&-2&0\\0&1&0\\0&0&1/\sqrt6\end{pmatrix}}
$$
sets $x_1+2x_2=y_1$, $x_2=y_2$, $\sqrt6x_3=y_3$, giving $Q_1=y_1^2+y_2^2+y_3^2$. The same transformation gives
$$
\boxed{Q_2=y_1^2+y_2^2-y_3^2.}
$$
Its signature is $(2,1)$. By <Sylvester's law of inertia> no real invertible change of variables can turn this second form into a sum of three positive squares.
Back to article page
