Please help me fill the step by step blanks to help me understand how SVD works thanks

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Please help me fill the step by step blanks to help me
understand how SVD works thanks

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We will work through these problems to introduce us to singular value decomposition (SVD). First we note that for any matrix Amxn, the matrices AT A and AAT are both symmetric. Singular Value Decomposition: Any m by n matrix A can be factored into A = UEVT = (orthogonal)(diagonal)(orthogonal) U is an m by m orthogonal matrix whose columns are the unit eigenvectors of AAT. is an n by n matrix orthogonal whose columns are the unit eigenvectors of AT A. = = D Σ = =(. ) is an m by n diagonal matrix. Let rank A = r < min(m, n). The matrix D is an r byr diagonal matrix whose diagonal is the vector of [01 02 or] ... The elements oi, 1<i <r are called singular values of A and they are the square roots of the r nonzero eigenvalues of AT A and AAT. The nonzero eigenvalues of AT A and AAT are ordered as 11 2 12 2 ... 22 and 0; = vi, 1 sisr.
ſo i 1 1 Problem 1: Let A We want to find the 1 SVD for A. First Compute AT A and AAT. 1. The size of AT A is by 2. The size of AAT is by We need to find the nonzero eigenvalues of AAT and AT A. Since they have the same eigenvalues we use the matrix with the smaller dimension. 3. The largest nonzero eigenvalue of AAT is 2, = 4. The smallest nonzero eigenvalue of AAT is 12 =
0 D= and 02 01 0 0 Σ = = 0 0 02 5. 01 = 6.02 = U = [ui u2] = \u21 U11 U12 where uy is the unit U22 eigenvector corresponding to 11 and u2 is the unit eigenvector corresponding to 12. 7. u11 = 8. U12 = 9. U21 = 10. U22
U11 = U12 U13 V = (v1 V2 V3] where V1 [ U21 U22 U23 U31 U32 U33 is the unit eigenvector corresponding to 11, V2 is the unit eigenvector corresponding to 12 and V3 is the unit eigenvector corresponding to 13 = 0. 11. U11 = 12. V12 = 13. U13 =
14. V21 = 15. U22 = 16. U23 = 17. 031 = 18. U32 = 19. U33 = Multiply the three matrices UCVT to verify that Α = UΣ/Τ.