(Solution) - Let A be a nonsingular 2 2 matrix with -(2025 Original AI-Free Solution)

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Let A be a nonsingular 2 × 2 matrix with singular value decomposition A = P ? QT and singular values ?1 ? ?2 > 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 ?1, ?2.
(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 ? |det A|.
(d) Find the principal axes, semi-axes, and area of the ellipses defined by
(i)