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-1has a solution . Let AERwhere m 2 n, and A has full rank. Show that...
3. Let A e IRmxn where where x minimizes llAz-bl 2. m 2 n, and A has full rank. Show that A = I a...
3. Let A e IRmxn where where x minimizes llAz-bl 2. m 2 n, and A has full rank. Show that A = I al has a solution wherez minimizes nAm2n2and Ahasfullrank, Showthatト2][2]-uhasasolution
3. Let A e IRmxn where where x minimizes llAz-bl 2. m 2 n, and A has full rank. Show that A = I al has a solution wherez minimizes nAm2n2and Ahasfullrank, Showthatト2][2]-uhasasolution
2. (5 pts) Assume A E Rm** with m > n has (full) rank n. Show that At = (ATA)TAT, What is the pse...
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...
3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12. 3. (15 pts.) Let A e Rmxn be a full rank matrix, m >...
3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12.
3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12.
6. Suppose A E Rnxm has full rank, that is, rank(A) min(n, n). Let ơi > > Ơr be the singular values of A. Let...
6. Suppose A E Rnxm has full rank, that is, rank(A) min(n, n). Let ơi > > Ơr be the singular values of A. Let B E Rnxm satisfy IA-B 2 < σ'. Then B also has full rank. Suppose A E Rnx'n has full rank, that is, rank(A)-r-min(n, n). Let ơi > > ơr be the singular values of A. Let B E Rnxm satisfy IIA-Blla < ơr. Then B also has full rank
6. Suppose A E Rnxm...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
9. (2 pts per part) Let A be an m x n matrix, where m >...
9. (2 pts per part) Let A be an m x n matrix, where m > n, and suppose that the rank of A is n (i.e., A has full column rank). Briefly justify your answers to each question below. a. Which two of the following statements are true? i. There are no vectors in Nul(A). ii. There is no basis for Nul(A). iii. dim(Nul(A)) = 0 iv. dim(Nul(A)) = m – n b. Are the columns of A a...
12.3 Least angle property of least squares. Suppose the m × n matrix A has linearly independent columns, and b is an m-vector. Let x ATb denote the least squares approximate solution (a) Show that fo...
12.3 Least angle property of least squares. Suppose the m × n matrix A has linearly independent columns, and b is an m-vector. Let x ATb denote the least squares approximate solution (a) Show that for any n-vector a, (Ax)Tb - (Aa)"(Aâ), i.e., the inner product of Ax and b is the same as the inner product of Ax and Ai. Hint. Use (Ax)b (ATb) and (ATA)2 = ATb (b) Show that when A and b are both nonzero, we...
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its rank. ii....
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its rank. ii. In case Z is just the n x 1 unit vector, i.e. Z- (1,....1)', what form does the vector Mz take? Note that x is any n- dimensional column vector
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its...
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and onl...
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0, then rank(A) + rank(B) 〈 n.
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0,...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), b...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
3. Let A e IRmxn where where x minimizes llAz-bl 2. m 2 n, and A has full rank. Show that A = I al has a solution wherez minimizes nAm2n2and Ahasfullrank, Showthatト2][2]-uhasasolution
3. Let A e IRmxn where where x minimizes llAz-bl 2. m 2 n, and A has full rank. Show that A = I al has a solution wherez minimizes nAm2n2and Ahasfullrank, Showthatト2][2]-uhasasolution
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...
3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12.
3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12.
6. Suppose A E Rnxm has full rank, that is, rank(A) min(n, n). Let ơi > > Ơr be the singular values of A. Let B E Rnxm satisfy IA-B 2 < σ'. Then B also has full rank. Suppose A E Rnx'n has full rank, that is, rank(A)-r-min(n, n). Let ơi > > ơr be the singular values of A. Let B E Rnxm satisfy IIA-Blla < ơr. Then B also has full rank
6. Suppose A E Rnxm...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
9. (2 pts per part) Let A be an m x n matrix, where m > n, and suppose that the rank of A is n (i.e., A has full column rank). Briefly justify your answers to each question below. a. Which two of the following statements are true? i. There are no vectors in Nul(A). ii. There is no basis for Nul(A). iii. dim(Nul(A)) = 0 iv. dim(Nul(A)) = m – n b. Are the columns of A a...
12.3 Least angle property of least squares. Suppose the m × n matrix A has linearly independent columns, and b is an m-vector. Let x ATb denote the least squares approximate solution (a) Show that for any n-vector a, (Ax)Tb - (Aa)"(Aâ), i.e., the inner product of Ax and b is the same as the inner product of Ax and Ai. Hint. Use (Ax)b (ATb) and (ATA)2 = ATb (b) Show that when A and b are both nonzero, we...
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its rank. ii. In case Z is just the n x 1 unit vector, i.e. Z- (1,....1)', what form does the vector Mz take? Note that x is any n- dimensional column vector
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its...
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0, then rank(A) + rank(B) 〈 n.
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0,...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
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