Solve x^2+xy+y^2=1: Find Rotation & Angle

[stunner5000pt]

Consider the equation x^2 + xy + y^2 = 1. Find a rotation so that the equation has no cross term.

here a=b=c=1 right
but how do i find an angle by which to rotate?
the text says the angle is pi/4... i don't know how they got that .. though

 

[Galileo]

Start by writing it as a matrix equation:

$$x^2+xy+y^2=\vec x^T A \vec x$$

 

[HallsofIvy]

Once you have the A Galileo is talking about, find its eigenvalues and eigenvectors. The eigenvectors give the directions for the new axes.