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Homework problem: Singular Value Decomposition Let A E R n 2 mn. Consider the singular value...
5. Let A 2 Rm£n. Show that (a) kerA = kerAtA; (b) rankAtA = rankAAt = rankA; (c) AtA and AAt have the same nonzero eigenvalues. Hint: Keep in mind the Singular Value Decomposition of matrices.
5. Let A 2 Rm£n. Show that (a) kerA = kerAtA; (b) rankAtA = rankAAt = rankA; (c) AtA and AAt have the same nonzero eigenvalues. Hint: Keep in mind the Singular Value Decomposition of matrices.
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...
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...
Let A E Mn(R) be a non-singular matrix. Show that if λ 1/λ is an eigenvalue...
Let A E Mn(R) be a non-singular matrix. Show that if λ 1/λ is an eigenvalue of A-1 0 is an eigenvalue of A, then
(4) The following is the singular value decomposition of a 3 x 4 matrix A with...
(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...
For the 3×2 matrix A: a) Determine the eigenvalues of ATA, and confirm that your eigenvalues...
For the 3×2 matrix A:
a) Determine the eigenvalues of ATA, and confirm that
your eigenvalues are consistent with the trace and determinant of
ATA.
b) Find an eigenvector for each eigenvalue of
ATA.
c) Find an invertible matrix P and a diagonal matrix D such that
P-1(ATA)P = D.
d) Find the singular value decomposition of the matrix A; that
is, find matrices U, Σ, and V such that A = UΣVT.
e) What is the best rank 1...
Find the singular value decomposition of A = [ (3 2 2), (2 3 2) ]...
Find the singular value decomposition of A = [ (3 2 2), (2 3 2) ] and determine the angle of rotation induced by U and V . Also, write the rank 1 decomposition of A in terms of the columns of U and rows of V . Can we do dimensionality reduction in this case? how to find angle of rotation induced by U and V? please provide the detailed process for above.
Q4. Let 1.01 0.99 0.99 0.98 (a) Find the eigenvalue decomposition of A. Recall that λ is an eigenvalue of A if for some u1],u2 (entries of the corresponding eigenvector) we have (1.01 u0.99u20 99...
Q4. Let 1.01 0.99 0.99 0.98 (a) Find the eigenvalue decomposition of A. Recall that λ is an eigenvalue of A if for some u1],u2 (entries of the corresponding eigenvector) we have (1.01 u0.99u20 99u [1] + (0.98-A)u[2] = 0. Another way of saying this is that we want the values of λ such that A-λ| (where I is the 2 x 2 identity matrix) has a non-trivial null space there is a nonzero vector u such that (A-AI)u =...
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find...
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find a (b) Determine the pseudoinverse matrix At, expressing At single matrix. as a (c) Consider the equation ) Ax 1 = and find the least squares approximation x' with minimum norm
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find a (b) Determine the pseudoinverse matrix At, expressing At single matrix. as a (c) Consider the equation...
Prove Theorem 4.2.21. The Singular Value Decomposition. PROVE THAT IF MATRIX A element of R^n*n Theorem...
Prove Theorem 4.2.21. The Singular Value
Decomposition. PROVE THAT IF MATRIX A element of R^n*n
Theorem 4.2.21. Let A e Rnxn. Then ||A| Definition 4.2.2. On R" we will use the standard inner product (7.7) = .2.2015 j=1 | 7 ||2=1 Theorem 4.2.20. Let A € R"X". Then ||A||2 = 01. Proof: Let AE Rnxn and let Let A=USVT be an SVD of A. We have || A||2 = max || 17 || 2 = max, ||UEV17 || 2 =...
True or False? If A is an m × n matrix and SVT is a singular value decomposition of A, then a vector u in Rn that minimizes || Au-bl is VyUlb where ΣΤ 1s the same as matrix Σ with singular values ok r...
True or False?
If A is an m × n matrix and SVT is a singular value decomposition of A, then a vector u in Rn that minimizes || Au-bl is VyUlb where ΣΤ 1s the same as matrix Σ with singular values ok replaced with 1/0k. Answer: _
If A is an m × n matrix and SVT is a singular value decomposition of A, then a vector u in Rn that minimizes || Au-bl is VyUlb where ΣΤ...
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...
Let A E Mn(R) be a non-singular matrix. Show that if λ 1/λ is an eigenvalue of A-1 0 is an eigenvalue of A, then
(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...
For the 3×2 matrix A:
a) Determine the eigenvalues of ATA, and confirm that
your eigenvalues are consistent with the trace and determinant of
ATA.
b) Find an eigenvector for each eigenvalue of
ATA.
c) Find an invertible matrix P and a diagonal matrix D such that
P-1(ATA)P = D.
d) Find the singular value decomposition of the matrix A; that
is, find matrices U, Σ, and V such that A = UΣVT.
e) What is the best rank 1...
Q4. Let 1.01 0.99 0.99 0.98 (a) Find the eigenvalue decomposition of A. Recall that λ is an eigenvalue of A if for some u1],u2 (entries of the corresponding eigenvector) we have (1.01 u0.99u20 99u [1] + (0.98-A)u[2] = 0. Another way of saying this is that we want the values of λ such that A-λ| (where I is the 2 x 2 identity matrix) has a non-trivial null space there is a nonzero vector u such that (A-AI)u =...
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find a (b) Determine the pseudoinverse matrix At, expressing At single matrix. as a (c) Consider the equation ) Ax 1 = and find the least squares approximation x' with minimum norm
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find a (b) Determine the pseudoinverse matrix At, expressing At single matrix. as a (c) Consider the equation...
Prove Theorem 4.2.21. The Singular Value
Decomposition. PROVE THAT IF MATRIX A element of R^n*n
Theorem 4.2.21. Let A e Rnxn. Then ||A| Definition 4.2.2. On R" we will use the standard inner product (7.7) = .2.2015 j=1 | 7 ||2=1 Theorem 4.2.20. Let A € R"X". Then ||A||2 = 01. Proof: Let AE Rnxn and let Let A=USVT be an SVD of A. We have || A||2 = max || 17 || 2 = max, ||UEV17 || 2 =...
True or False?
If A is an m × n matrix and SVT is a singular value decomposition of A, then a vector u in Rn that minimizes || Au-bl is VyUlb where ΣΤ 1s the same as matrix Σ with singular values ok replaced with 1/0k. Answer: _
If A is an m × n matrix and SVT is a singular value decomposition of A, then a vector u in Rn that minimizes || Au-bl is VyUlb where ΣΤ...
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