Question

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8 0 5 (1 point) Find the singular values 01 2 o2 2 o3 of A -5 0 8 (1 point) A singular value decomposition of a matrix A is as follows: 15 0.5 0.5 0.5 0.5 0 0.8 0.5 -0.5 -0.5 0.5 0 5 0.6 A -0.5 0.5 -0.5 0.5 0 0 0.6 -0.8 -0.5 -0.5 0.5 0.5 0 1. Find the closest (with respect to the Frobenius norm) matrix of rank 1 to A A1 = 2. Find the Frobenius norm of A - A1 1|A A1||F (1 point) A singular value decomposition of A is as follows: 20 0.5 0.5 0.5 0.5 0 O 20 T0.8 -0.6 0.5 -0.5 0.5 -0.5 A=UΣVT 0.5 -0.5 -0.5 0.5 0 0 0.6 0.8 |0.5 Find the least-squares solution of the linear system -0.5 -0.5 0.5 0 0 -5 -2 Ax b, where b _ -1 1 LO (1 point) Find the singular values o1 2 02 of 2 A = 1 -1 -2 σ1 σ2 Find unit vectors v, and 2 such that || Aū1 || = o1 and || Av || = 02. 7 4 12 X Find an invertible matrix X and a diagonal matrix D such that D AX (1 point) Let A = -1 6 -3 10 X = D = Hint: The eigenvalues are 2,-1 and 1

Answer 1

The matrix is 0 5 8 A = -5 0 8 So, 8 5 A' = | 0 5 8 So, -5 89 5 0 AA 89 5 5 8 89 -5 8 0 5 AA=0 0 -5 0 8 0 5 89 So, the characterstic equation is (89-20 A 89,89 And, - (89-λ) -0 A = 0,89,89 So, the eigenvalues are A 0,89,89 So, the singular values are 89 89