Let A be a nonsingular 2 Ã 2 matrix with singular value decomposition A = P QT and singular values Ï1 ¥ Ï2 > 0.

Let A be a nonsingular 2 × 2 matrix with singular value decomposition A = P ˆ‘ QT and singular values σ1 ‰¥ σ2 > 0.
(a) Prove that the image of the unit (Euclidean) circle under the linear transformation defined by A is an ellipse. E = {Ax| ||x|| = 1}, whose principal axes are the columns p1, p2 of P. and whose corresponding semi-axes are the singular values σ1, σ2.
(b) Show that if A is symmetric, then the ellipse's principal axes are the eigenvectors of A and the semi-axes are the absolute values of its eigenvalues.
(c) Prove that the area of E equals π |det A|.
(d) Find the principal axes, semi-axes, and area of the ellipses defined by
(i)

(ii)

(iii)

(e) What happens if A is singular

(o 1) 0 1 5 -4 0 -3