= Problem 3. Suppose that A is m xn matrix with rank k, singular value decomposition A USVT, and reduced singular value decomposition A = USVT, where U = (u₁ u₂ um) and V (V₁ V2 V). Suppose that the eigenvalues of ATA satisfy A₁₂.. Σλπ. (a) For all i[n] (not just [k]), compute ||Av. What happens if i > k? (Hint: Look at your calculations for Problems 2a and 2b in Lab 3) = Aw (i.e. y (b) Let y Rm such that there is a wR" such that y Col(A)). Show that there is a zЄ R such that y = Üz (that is, show that y = Col(U)) (Hint: W = C₁V1 + C2V2+...+CVn for some constants C1, C2,..., Cn). (c) It follows from Lab 3 (Problem 2) and (b) above that Col(A) = Col(U). Show that UUTb = projcol(4) (b) (Hint: Use the "column-row expansion" of UUT). (d) Use (c) above to verify your answer from Problem 2 (c). Problem 2. Suppose that A is m x n matrix with rank k and a singular value decompo- sition AUSVT, where U = (u₁ u2 um) and V = (V₁ V2 Vn). Recall that V1, V2, V, are pairwise orthonormal eigenvectors of ATA with corresponding eigenvalues A1, A2, A and where σ = √ for all i [k]. -Avi, (a) Show that u₁, u2,..., u are all unit vectors. (b) Show that u, u2... u are pairwise orthogonal. (c) Show that u₁, U2, - , u, are eigenvectors of AAT. What are the corresponding eigen- values? (d) Find a singular value decomposition of AT. (e) Suppose that A is square (mn) and invertible. Find a singular value decomposition of A¹ (Hint: What do you know about the eigenvalues/singular values of A?). Avi Si and AAU = divi || U; 11² = <ui, u;> =< Av; Av;> <v, ATAU;> = <vi, div> Ai N <v, vi). A = √² 1. be be orthonamal Since {v,,, set of vectors. . {1, 2,..., Ux ) are all unit vectors. ⑥<U; U; > = (± Avi, & Au;> "j itj. V Si (u, ATAU 5:5j 6: [: {}, be orthonormal set] be pairwise orthogonal Set of vectors. Now, = Consider, AAU AC AAT Av Ji =AATAV Si = A divi Si = di Avi Si 7 di li di be eigen value of ui correspondy vector of matrix AAT. So values are A1, A2, A3, Як eigen 3 Singular decomposition of AT. A = USVT AT = √ STUT→ Singular decomposition of AT. (Au Az USVT A-(v) su be singular de composition of A".