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6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form,...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
Solve part (d) 6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the proble...
Solve part (d)
6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain (c) Is the operator S symmetric? Explain (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 ln x, 1 < 2, in terms of these (e) Find the orthogonal expansion of f(x) eigenfunctions _
6. Consider the eigenvalue problem 2"xy3y...
3. Consider the eigenvalue problem 1
The answer is given. Please show more detailed steps, thank
you.
3. Consider the eigenvalue problem 1<x<2 dx2 y(1)=0,y(2) = 0. dx iwrite it in the standard Sturm-Liouville form. ii) Show that 0 by the Rayleigh Quotient. dx p(x)-x, q(x) = 0, σ(x)-1 According the Raileigh Quotient Any eigenvalue is related to its eigenfunction φ(x) by - x p(x) dr Since the B.C. are ф(1)-0 and ф(2-0, so dx
3. Consider the eigenvalue problem 1
Problem 11 Let A= 3 -1 0 0 16 -5 0 0 16 0 -2 15...
Problem 11 Let A= 3 -1 0 0 16 -5 0 0 16 0 -2 15 -3 -15 2 8 a) [3 pts) Compute the characteristic polynomial of A and find its roots. b) (4 pts] For each eigenvalue of A find a basis for the corresponding eigenspace. c) [3 pts] Determine if A is defective. Justify your answer. d) [6 pts) If A is defective, determine the defective eigenvalue or eigenvalues, and find a Jordan chain (or set of...
3 -1 0 Problem 11 Let A= 16 -5 0 0 16 0 -2 15 -3...
3 -1 0 Problem 11 Let A= 16 -5 0 0 16 0 -2 15 -3 -15 2 8 0 a) [3 pts) Compute the characteristic polynomial of A and find its roots. b) [4 pts) For each eigenvalue of A find a basis for the corresponding eigenspace. c) [3 pts] Determine if A is defective. Justify your answer. d) [6 pts) If A is defective, determine the defective eigenvalue or eigenvalues, and find a Jordan chain (or set of...
Problem 2 Is λ 3 an eigenvalue of 13-2 / ? If so, find a corresponding...
Problem 2 Is λ 3 an eigenvalue of 13-2 / ? If so, find a corresponding eigenvector.
Verify that i; is an eigenvalue of A and that x; is a corresponding eigenvector. -1...
Verify that i; is an eigenvalue of A and that x; is a corresponding eigenvector. -1 4 -6 A = 4 5 -12 -2 - 4 3 11 = 13, X1 = (1, 2, -1) 12 = -3, X2 = (-2, 10) 13 = -3, X3 = (3, 0, 1) 1 AX1 5 -12 = 13 2 = 11X1 -1 Ax2 4 5 -12 -2 -4 3 22X2 -1 3 3 4 5 -6 - 12 Ах3 4 1: -3...
Question 5 [3+(2+4) marks] (a) The matrix A has a repeated eigenvalue of 1 = 2....
Question 5 [3+(2+4) marks] (a) The matrix A has a repeated eigenvalue of 1 = 2. During the solution of the solution (A-21)X = 0, the augmented matrix below appears. Find a basis for the eigenspace for this eigenvalue. Ti 0 -2 07 lo o o lo To ooo (b) (i) Show that if T(x) is a linear transformation from R" to R", that T(0) is the zero vector. (i) Assume that T(u) = 0 only when u = 0....
Is A=3 an eigenvalue of A. If so, find one corresponding eigenvector. -1 0-2 2 5...
Is A=3 an eigenvalue of A. If so, find one corresponding eigenvector. -1 0-2 2 5 - 4 0 2 -2 a. v=(-1,5,2) b. V=(1,5,1) c. V=(-5,6,1) d. X = 3 is not aneigenvlalue of A оа Ob ос
Verify that i; is an eigenvalue of A and that x; is a corresponding eigenvector. 5...
Verify that i; is an eigenvalue of A and that x; is a corresponding eigenvector. 5 -1 4 A = 0 3 1 21 = 5, X1 = (1, 0, 0) 12 = 3, X2 = (1, 2, 0) 13 = 4, X3 = (-3, 1, 1) 0 04 5 -1 4 1 1 Ax1 = 0 3 1 0 = 5 0 11 III = 11X1 0 04 0 0 5 -1 4 1 1 Ax2 0 3 1...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
Solve part (d)
6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain (c) Is the operator S symmetric? Explain (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 ln x, 1 < 2, in terms of these (e) Find the orthogonal expansion of f(x) eigenfunctions _
6. Consider the eigenvalue problem 2"xy3y...
The answer is given. Please show more detailed steps, thank
you.
3. Consider the eigenvalue problem 1<x<2 dx2 y(1)=0,y(2) = 0. dx iwrite it in the standard Sturm-Liouville form. ii) Show that 0 by the Rayleigh Quotient. dx p(x)-x, q(x) = 0, σ(x)-1 According the Raileigh Quotient Any eigenvalue is related to its eigenfunction φ(x) by - x p(x) dr Since the B.C. are ф(1)-0 and ф(2-0, so dx
3. Consider the eigenvalue problem 1
Problem 11 Let A= 3 -1 0 0 16 -5 0 0 16 0 -2 15 -3 -15 2 8 a) [3 pts) Compute the characteristic polynomial of A and find its roots. b) (4 pts] For each eigenvalue of A find a basis for the corresponding eigenspace. c) [3 pts] Determine if A is defective. Justify your answer. d) [6 pts) If A is defective, determine the defective eigenvalue or eigenvalues, and find a Jordan chain (or set of...
3 -1 0 Problem 11 Let A= 16 -5 0 0 16 0 -2 15 -3 -15 2 8 0 a) [3 pts) Compute the characteristic polynomial of A and find its roots. b) [4 pts) For each eigenvalue of A find a basis for the corresponding eigenspace. c) [3 pts] Determine if A is defective. Justify your answer. d) [6 pts) If A is defective, determine the defective eigenvalue or eigenvalues, and find a Jordan chain (or set of...
Problem 2 Is λ 3 an eigenvalue of 13-2 / ? If so, find a corresponding eigenvector.
Verify that i; is an eigenvalue of A and that x; is a corresponding eigenvector. -1 4 -6 A = 4 5 -12 -2 - 4 3 11 = 13, X1 = (1, 2, -1) 12 = -3, X2 = (-2, 10) 13 = -3, X3 = (3, 0, 1) 1 AX1 5 -12 = 13 2 = 11X1 -1 Ax2 4 5 -12 -2 -4 3 22X2 -1 3 3 4 5 -6 - 12 Ах3 4 1: -3...
Question 5 [3+(2+4) marks] (a) The matrix A has a repeated eigenvalue of 1 = 2. During the solution of the solution (A-21)X = 0, the augmented matrix below appears. Find a basis for the eigenspace for this eigenvalue. Ti 0 -2 07 lo o o lo To ooo (b) (i) Show that if T(x) is a linear transformation from R" to R", that T(0) is the zero vector. (i) Assume that T(u) = 0 only when u = 0....
Is A=3 an eigenvalue of A. If so, find one corresponding eigenvector. -1 0-2 2 5 - 4 0 2 -2 a. v=(-1,5,2) b. V=(1,5,1) c. V=(-5,6,1) d. X = 3 is not aneigenvlalue of A оа Ob ос
Verify that i; is an eigenvalue of A and that x; is a corresponding eigenvector. 5 -1 4 A = 0 3 1 21 = 5, X1 = (1, 0, 0) 12 = 3, X2 = (1, 2, 0) 13 = 4, X3 = (-3, 1, 1) 0 04 5 -1 4 1 1 Ax1 = 0 3 1 0 = 5 0 11 III = 11X1 0 04 0 0 5 -1 4 1 1 Ax2 0 3 1...
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