Answer Happy

3. (15 points) Find the singular value decomposition of the following matrix A=7] -UDVT It turns out that V contains the eigenvectors of AT A and U contains the eigenvectors of AAT. The columns of U can be easily computed using Avi U₂ = In practice, you would choose the smaller of the two AA and . σi AT A, as one can be computed from the other.
(a) Find the eigenvalues of A, and eigenvectors v, of AT A. Make sure these vectors are orthonormal. Since ATA is a positive operator, the eigenvectors are orthogonal, so just turn them into unit vectors by dividing its norm. (b) The matrix D contains the values ;=√₁. The values o, are called the singular values of A. The number of nonzero singular values is equal to the rank of A. (c) Compute the eigenvectors u, of AAT using the formula given above. (d) Verify that A = UDVT.