Let K be a positive definite 2 × 2 matrix.
(a) Explain why the quadratic equation xT Kx = 1 defines an ellipse. Prove that its principal axes are the eigenvectors of K, and the semi-axes are the reciprocals of the square roots of the eigenvalues.
(b) Graph and describe the following curves:
(i) x2 + 4y2 = 1
(ii) x2 + xy + y2 = 1
(iii) 3x2 + 2xy + y2 = 1
(c) What sort of curve(s) does xT K x = 1 describe if K is not positive definite?