Past exam of the mathematical methods I course of the University of Oxford / 2015 / 6 / Solution

The characteristic polynomial is $(5-\lambda )(3-\lambda )-3=(\lambda -6)(\lambda -2)$. Corresponding normalized eigenvectors areThese form an orthonormal basis. Write $x=u{e}_6+v{e}_2$; the quadratic form becomesThe locus is an ellipse centred at the origin. Its minor semiaxis has length $\sqrt{2/3}$ along the direction $30^{\circ }$ from the positive horizontal axis; its major semiaxis has length $\sqrt{2}$ along $120^{\circ }$. The right panel of the figure in part 1 shows these principal directions.