Any symmetric diagonalizable real valued matrix A in R* admits an eigenvalue decomposition of the following...

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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, Σ € Rmx and V E Rnxn and U, V have their columns to be orthonormal. The matrix Σ € 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, Σ 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 = ΣAuu, (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). 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, Σ € Rmx and V E Rnxn and U, V have their columns to be orthonormal. The matrix Σ € 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, Σ 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 = ΣAuu, (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).

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a Singular Value Decomposition SVD We want to find the SVD of the given matrices A The SVD consists ... View the full answer