4. (Heath 4.3) Given an approximate eigenvector x of A, what is the best estimate (in...
4. (Heath 4.3) Given an approximate eigenvector x of A, what is the best estimate (in the least squares sense) for the corresponding eigenvalue?
Homework Answers
Rayleigh Quotient
Given an approximate eigenvector x of A, the Rayleigh Quotient is the best estimate.
Consider the eigenvalue equation xλ = Ax where x is our current best eigenvector guess and λ is unknown. the coefficient matrix is the n × 1 matrix x. The normal equations say that the least squares answer is the solution of x^ T xλ = x ^T Ax, or λ = (x^T Ax) / (x ^T x) . λ is known as the Rayleigh Quotient.
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4. (Heath 4.3) Given an approximate eigenvector x of A, what is
the best estimate (in...
For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue. -4 A =...
For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue. -4 A = X = 5 48-11
Given the matrix A= 76 -2 -4 -4 8 8 1 4 -4 -4 X =...
Given the matrix A= 76 -2 -4 -4 8 8 1 4 -4 -4 X = 2 is an eigenvalue of A and 12 = 4 is an eigenvalue of A of multiplicity 2. (a) Find the eigenvector(s) corresponding to l1 = 2. (b) Find the eigenvector(s) corresponding to 12 = 4. (C) Find the general solution of x' = Ax.
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
Given the matrix A467 333 0 2 0 0 The eigenvector associated with the largest eigenvalue...
Given the matrix A467 333 0 2 0 0 The eigenvector associated with the largest eigenvalue ofA is [7-3 1]T (a) Determine the eigenvalue associated with this eigenvector (b) For b-[1 1 1]T, find the approximate solution to b in the system x-Ab due to this eigenvector and compare it the exact solution.
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c...
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c be a real number. Show that v is an eigenvector of A+cI, where I is the appropriately sized identity matrix. What is the corresponding eigenvalue?
0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue 0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1...
0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue
0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue
Verify that ; is an eigenvalue of A and that x; is a corresponding eigenvector. A...
Verify that ; is an eigenvalue of A and that x; is a corresponding eigenvector. A = [3 0 ] LO-3 14 = 3, x1 = (1, 0) 12 = -3, X2 = (0, 1) AX1 = = 3 = 11X1 10 -3 1 0 **(4- = -1) --- --6-418- | | |--[:)--- Ax2 = = -3 = 12x2 -3
3-5a 8. Let A 2 0 1.I It is given that 0 is an eigenvector for...
3-5a 8. Let A 2 0 1.I It is given that 0 is an eigenvector for 2 -3 7 (a) What is the corresponding eigenvalue? (b) What is the value of a?
3. For matrix 2 2 3 x Power me+hod A 2 4 5 L3 5 7 use the power method to estimate the eigenvalue of greatest absolute value and a malized eigenvector. Note that I'm not asking what Wolfram A...
3. For matrix 2 2 3 x Power me+hod A 2 4 5 L3 5 7 use the power method to estimate the eigenvalue of greatest absolute value and a malized eigenvector. Note that I'm not asking what Wolfram Alpha or Matlab or whatever says the answer is. I want to know how the power method acts. Does it converge quickly? Slowly? Not at all?
3. For matrix 2 2 3 x Power me+hod A 2 4 5 L3 5...
13. Use three iterations of the power method to estimate the largest eigenvalue and corresponding eigenvector...
13. Use three iterations of the power method to estimate the largest eigenvalue and corresponding eigenvector of A-2 4 to help with the arithmetic. Compare your estimates to the true values. For full credit, you must show all of your work and report each of the intermediate estimates xi, x2,A1, ?2 as well as the final estimates x3 and 23 ,starting with xo and ending with x3. You may use Matlab 0
For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue. -4 A = X = 5 48-11
Given the matrix A= 76 -2 -4 -4 8 8 1 4 -4 -4 X = 2 is an eigenvalue of A and 12 = 4 is an eigenvalue of A of multiplicity 2. (a) Find the eigenvector(s) corresponding to l1 = 2. (b) Find the eigenvector(s) corresponding to 12 = 4. (C) Find the general solution of x' = Ax.
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
Given the matrix A467 333 0 2 0 0 The eigenvector associated with the largest eigenvalue ofA is [7-3 1]T (a) Determine the eigenvalue associated with this eigenvector (b) For b-[1 1 1]T, find the approximate solution to b in the system x-Ab due to this eigenvector and compare it the exact solution.
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c be a real number. Show that v is an eigenvector of A+cI, where I is the appropriately sized identity matrix. What is the corresponding eigenvalue?
0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue
0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue
Verify that ; is an eigenvalue of A and that x; is a corresponding eigenvector. A = [3 0 ] LO-3 14 = 3, x1 = (1, 0) 12 = -3, X2 = (0, 1) AX1 = = 3 = 11X1 10 -3 1 0 **(4- = -1) --- --6-418- | | |--[:)--- Ax2 = = -3 = 12x2 -3
3-5a 8. Let A 2 0 1.I It is given that 0 is an eigenvector for 2 -3 7 (a) What is the corresponding eigenvalue? (b) What is the value of a?
3. For matrix 2 2 3 x Power me+hod A 2 4 5 L3 5 7 use the power method to estimate the eigenvalue of greatest absolute value and a malized eigenvector. Note that I'm not asking what Wolfram Alpha or Matlab or whatever says the answer is. I want to know how the power method acts. Does it converge quickly? Slowly? Not at all?
3. For matrix 2 2 3 x Power me+hod A 2 4 5 L3 5...
13. Use three iterations of the power method to estimate the largest eigenvalue and corresponding eigenvector of A-2 4 to help with the arithmetic. Compare your estimates to the true values. For full credit, you must show all of your work and report each of the intermediate estimates xi, x2,A1, ?2 as well as the final estimates x3 and 23 ,starting with xo and ending with x3. You may use Matlab 0
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