Solve the following problem.

Problem 1(*): Consider the matrix 7 1 A = 0 -12 14 10 . Compute the matrix ATA . What does the spectral theorem say about ATA? . Find the eigenvectors and eigenvalues of ATA. . Find an orthogonal basis {u1, u2} of eigenvectors of ATA. . Show that the vectors Au, and Au2 are orthogonal eigenvectors of AAT. What are their magnitudes? . Normalize the vectors Au, and Aug to obtain orthonormal eigenvectors of and 12 of AAT. . Verify that A = oiviu] + 02v242. Here of and o? are the eigenvalues of AT A