Any symmetric diagonalizable real valued matrix A in R* admits an eigenvalue decomposition of the following form: A = - UAUT, where the columns of U are orthonormal. This leads to a decomposition of the following form: i=1 where A, is the i-th eigenvalue of A and u, is the corresponding eigenvector. Note that, each term in the summation above is a rank 1 matrix. We would like to extend this form of spectral decomposition to arbitrary rectangular matrices as well. Technically speaking, given a matrix A Rmx, we would like to decompose it in the following way: A = UEVT, where U Rmxm, Rmx and V E Rnxn and U, V have their columns to be orthonormal. The matrix Rmx is a diagonal matrix, i.e., all elements except the main diagonal are 0. (a) Find the singular value decomposition of the following matrices, i.e., compute the matrices U, and V. We will do it in the following steps: (10) Is AA diagonalizable? If so, find out the matrix U by diagonalizing AA. Is ATA diagonalizable? If so, find out the matrix V by diagonalizing ATA. Find E. Use these steps for this question. - (110) (i) A = 11 A = Auu, (ii) A = 1 1 10 3 5 0 6 1 (b) Compute the singular values of the following matrices. (i) A= -1 0 2 -1 -1 (5) (5) (10) (5) (5) (ii) A = (c) If A is symmetric, what is the relationship between the eigenvalues and singular values? (5) Hint: You may need to use MATLAB or Python to compute the eigenvalues and eigenvectors in parts (a) and (b).