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3. Eigenvalues. Consider the symmetric matrix 3 1 0 A=112 0 1 4/5 (a) How many eigenvalues of A are between 3 and 0? (Don't explicitly compute them. It will take too long.) (b) Is A positive...
F GHANA served) TEF RO HC 3 -2 0 A2. Given the matrix below 5 marks) [5 marks (10 marks (b) Compute explicitly the eige...
F GHANA served) TEF RO HC 3 -2 0 A2. Given the matrix below 5 marks) [5 marks (10 marks (b) Compute explicitly the eigenvalues and determine the determinant, (c) Compute the corresponding eigenvectors of the matrix above (a) Show that the matrix is positive definite. 1 | so that the characteristic polynomial 5 marks 0 (d) Choose a, band c in the matrix B = | 0 Based on Cayley-Hamilton's theorem, every matrix fulfills its characteristic polynomial, using the...
A real symmetric matrix B e Rnxn (i.e. BT = B) is said to be positive...
A real symmetric matrix B e Rnxn (i.e. BT = B) is said to be positive definite if all of its eigenvalues 11, 12, ..., In are positive. (Recall that is an eigenvalue of B if and only if there exits a nonzero vector t such that Bt = it). Show that B-1 is also positive definite. That is, you need to show that all the eigenvalues of B-1 are also positive. (Hint: consider equation Bt; = liti for all...
2a. Given a linear system of equations A b with a symmetric positive definite matrix A ERIX4 whic...
2a. Given a linear system of equations A b with a symmetric positive definite matrix A ERIX4 which has eigenvalues 1, 1/4, 1/9,1/16. Consider the iterative method defined by r(k +1) = r(k)-w(Ax(k)-b). Can you choose w such that method is convergent? If so, what is the best possible w? 2b. Discuss the convergence of the Jacobi method for Ar-b with the tridiagonal matrix -1 3 Does the Jacobi method converge for this matrix? What is the convergence rate
2a....
# 2: Consider the real symmetric matrix A= 4 1 a) What are the eigenvalues and...
# 2: Consider the real symmetric matrix A= 4 1 a) What are the eigenvalues and eigenvectors. [Hint: Use wolframalpha.] b) What is the trace of A, what is the sum of the eigenvalues of A. What is a general theorem th c) The eigenvalues of A are real. What is a general theorem which assert conditions that t d) Check that the eigenvectors are real. What is a general theorem which asserts conditions th asserts equality? eigenvalues are real...
Let 4-β 0 0 A=1 0 4-3 024-β where β > 0 is a parameter. (a) Find the eigenvalues of A (note the eigenvalues will be functions of β). (b) Determine the values of β for which the matrix A is positiv...
Let 4-β 0 0 A=1 0 4-3 024-β where β > 0 is a parameter. (a) Find the eigenvalues of A (note the eigenvalues will be functions of β). (b) Determine the values of β for which the matrix A is positive definite. Determine the values of β for which the matrix A is positive semidefinite. (c) For each eigenvalue of A, find a basis for the corresponding eigenspace. (d) Find an orthonormal basis for R3 consisting of eigenvectors of...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix...
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix B of # 3. 5 1 # 4 a) From the matrix fact sheet, what are the eigenvalues of B and its characteristic polynomial ? # 4 b) Compute the trace of B, the sum of diagonal terms of the matrix B in three ways. 1- directly, 2- via eigenvalues, 3-via characteristic polynomial # 4 c) Compute the determinant of B, ad — bc,...
5. Recall that a symmetric matrix A is positive definite (SPD for short) if and only...
5. Recall that a symmetric matrix A is positive definite (SPD for short) if and only if T Ar > O for every nonzero vector 2. 5a. Find a 2-by-2 matrix A that (1) is symmetric, (2) is not singular, and (3) has all its elements greater than zero, but (1) is not SPD. Show a nonzero vector such that zAx < 0. 5b. Let B be a nonsingular matrix, of any size, not necessarily symmetric. Prove that the matrix...
3 1. Let A = 0 (a) Compute the eigenvalues of A and specify their algebraic...
3 1. Let A = 0 (a) Compute the eigenvalues of A and specify their algebraic multiplicities. (b) For every eigenvalue 1, determine the eigenspace Ex and specify its dimension. (c) Is A a defective matrix? Why or why not? (d) Is A a singular matrix? Why or why not? (e) Determine the eigenvalues of (74) + 5.
Q2. Consider the matrix A 6 3 0 -1 0-2 0 5 (a) Find all eigenvalues...
Q2. Consider the matrix A 6 3 0 -1 0-2 0 5 (a) Find all eigenvalues of the matrix A. (b) Find all eigenvectors of the matrix A. (c) Do you think that the set of the eigenvectors of A is a basis for the vector space R3? (Justify your answer
F GHANA served) TEF RO HC 3 -2 0 A2. Given the matrix below 5 marks) [5 marks (10 marks (b) Compute explicitly the eigenvalues and determine the determinant, (c) Compute the corresponding eigenvectors of the matrix above (a) Show that the matrix is positive definite. 1 | so that the characteristic polynomial 5 marks 0 (d) Choose a, band c in the matrix B = | 0 Based on Cayley-Hamilton's theorem, every matrix fulfills its characteristic polynomial, using the...
A real symmetric matrix B e Rnxn (i.e. BT = B) is said to be positive definite if all of its eigenvalues 11, 12, ..., In are positive. (Recall that is an eigenvalue of B if and only if there exits a nonzero vector t such that Bt = it). Show that B-1 is also positive definite. That is, you need to show that all the eigenvalues of B-1 are also positive. (Hint: consider equation Bt; = liti for all...
2a. Given a linear system of equations A b with a symmetric positive definite matrix A ERIX4 which has eigenvalues 1, 1/4, 1/9,1/16. Consider the iterative method defined by r(k +1) = r(k)-w(Ax(k)-b). Can you choose w such that method is convergent? If so, what is the best possible w? 2b. Discuss the convergence of the Jacobi method for Ar-b with the tridiagonal matrix -1 3 Does the Jacobi method converge for this matrix? What is the convergence rate
2a....
# 2: Consider the real symmetric matrix A= 4 1 a) What are the eigenvalues and eigenvectors. [Hint: Use wolframalpha.] b) What is the trace of A, what is the sum of the eigenvalues of A. What is a general theorem th c) The eigenvalues of A are real. What is a general theorem which assert conditions that t d) Check that the eigenvectors are real. What is a general theorem which asserts conditions th asserts equality? eigenvalues are real...
Let 4-β 0 0 A=1 0 4-3 024-β where β > 0 is a parameter. (a) Find the eigenvalues of A (note the eigenvalues will be functions of β). (b) Determine the values of β for which the matrix A is positive definite. Determine the values of β for which the matrix A is positive semidefinite. (c) For each eigenvalue of A, find a basis for the corresponding eigenspace. (d) Find an orthonormal basis for R3 consisting of eigenvectors of...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix B of # 3. 5 1 # 4 a) From the matrix fact sheet, what are the eigenvalues of B and its characteristic polynomial ? # 4 b) Compute the trace of B, the sum of diagonal terms of the matrix B in three ways. 1- directly, 2- via eigenvalues, 3-via characteristic polynomial # 4 c) Compute the determinant of B, ad — bc,...
5. Recall that a symmetric matrix A is positive definite (SPD for short) if and only if T Ar > O for every nonzero vector 2. 5a. Find a 2-by-2 matrix A that (1) is symmetric, (2) is not singular, and (3) has all its elements greater than zero, but (1) is not SPD. Show a nonzero vector such that zAx < 0. 5b. Let B be a nonsingular matrix, of any size, not necessarily symmetric. Prove that the matrix...
3 1. Let A = 0 (a) Compute the eigenvalues of A and specify their algebraic multiplicities. (b) For every eigenvalue 1, determine the eigenspace Ex and specify its dimension. (c) Is A a defective matrix? Why or why not? (d) Is A a singular matrix? Why or why not? (e) Determine the eigenvalues of (74) + 5.
Q2. Consider the matrix A 6 3 0 -1 0-2 0 5 (a) Find all eigenvalues of the matrix A. (b) Find all eigenvectors of the matrix A. (c) Do you think that the set of the eigenvectors of A is a basis for the vector space R3? (Justify your answer
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