Problem 29: Find a 2-by-3 matrix having rank 1 whose singular value is 2, left singular vector is (1,2)7/V5, and right singular vector is (1,0,1)7/v2. Problem 29: Find a 2-by-3 matrix having ran...
(4) The following is the singular value decomposition of a 3 x 4 matrix A with some entries not given 1/3 -2/V5 1/v5 2/3 2/3 3 0 12/13 5/13 3/5 4/5 5/13 12/13 0 0 A 0 2 0 0 0 0 0 0 0 (a) What are the eigenvalues of AAT? of ATA? What is the rank of A? 1 2 (b) Find a non-zero vector w such that AAT = 9w. such that ATAu 4u. (c) Find a...
True or False? 1. If σ is a singular value of a matrix A, then σ is an eigenvalue of ATA Answer: 2. Every matrix has the same singular values as its transpose Answer: 3. A matrix has a pseudo-inverse if and only if it is not invertible. Answer: 4. If matrix A has rank k, then A has k singular values Answer:_ 5. Every matrix has a singular value decomposit ion Answer:_ 6. Every matrix has a unique singular...
8. Consider the real matrix As -1 0 (a) (3 pts) Find the singular values of A. (b) (4 pts) Find a singular value decomposition of A. (c) (3 pts) Find 8. Consider the real matrix As -1 0 (a) (3 pts) Find the singular values of A. (b) (4 pts) Find a singular value decomposition of A. (c) (3 pts) Find
(1) Let u = (-1,2) and v = (3, 1). (a) (5] Find graphically the vector w = (2u - v). (b) (5] Find algebraically the vector z=3u - 2 (2) (a) [5] Write u ='(1, -5, -1) as a linear combination of v1 = (1,2,0), v2 = (0,1,-1), V3 = (2,1,1). (b) (5] Are the 4 vectors u, V1, V2, V3 linearly independent? Explain your answer. (C) (5) Are the 2 vectors V, V3 linearly independent? Explain your answer....
2. (5 pts) Assume A E Rm** with m > n has (full) rank n. Show that At = (ATA)TAT, What is the pseudo-inverse of a vector u R" regarded as an m x 1 matrix? 3. (5 pts) Let B AT where A is the matrix in Problem 1. Use Matlab to find the singular value decomposition and the Moore-Penrose pseudo-inverse of B. Then solve minimum-norm least squares problem minl-ll : FE R minimizes IBr-ey where c- [1,2. Compare...
=[e - ] f,l which depends 3. (5 points each-10 points) Consider the rank one matrix A- on two real parameters e and f. (a) Find the singular value decomposition (SVD) A-υΣγί =[e - ] f,l which depends 3. (5 points each-10 points) Consider the rank one matrix A- on two real parameters e and f. (a) Find the singular value decomposition (SVD) A-υΣγί
(20) Let W be spanned by (1,1,0)7 and (1,-1,2)T in R3x1 Find the projection matrix from R3x1 onto W (a) (1,1,1)7 in W? (b) Is the vector b (c) Find the solutionx to the least square problem for Ax = b. (d) What is the vector in W that best approximates b? (20) Let W be spanned by (1,1,0)7 and (1,-1,2)T in R3x1 Find the projection matrix from R3x1 onto W (a) (1,1,1)7 in W? (b) Is the vector b...
Find all values of k that make the fol- Problem 7 lowing matrix singular: [i 20 k] 1 1 k k2 1 0 1 k 0 1 1 2
A = 1. Perform singular value decomposition. 2. Find the pseudo inverse of A. 3. Obtain a set of vector x that minimizes 1 1 1 1 0 0 | Ax (1,2,3)7 1 1 1 1 0 0 | Ax (1,2,3)7
Find an orthogonal basis for the column space of the matrix to the right. -1 5 5 1 -7 4 1 - 1 7 1 -3 -4 An orthogonal basis for the column space of the given matrix is O. (Type a vector or list of vectors. Use a comma to separate vectors as needed.) The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for 3 W. 6 -2 An...