Pure-quadratic polynomial/R/2 variables/Standard form/1/Example

We consider the pure-quadratic polynomial

$F=3X^2-4XY+5Y^2.$

In order to apply fact, we have to find the eigenvalues of the matrix

$M=\left(\begin{array}{cc}3& -2\\ -2& 5\end{array}\right).$

The characteristic polynomial is

$(X-3)(X-5)-4=X^2-8X+11=(X-4{)}^2-5.$

Therefore, the eigenvalues are

$x_1=\sqrt{5}+4\text{ and }{x}_2=-\sqrt{5}+4.$

Therefore, in a suitable orthonormal basis, the polynomial has the form

$F=(\sqrt{5}+4)U^2+(-\sqrt{5}+4)V^2.$