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The symmetric matrix A below has eigenvalues 15 and -15 (multiplicity 2). Find an orthonormal basis...
The symmetric matrix A below has distinct eigenvalues 10,-2 and-8. Find an orthogonal matrix P and...
The symmetric matrix A below has distinct eigenvalues 10,-2 and-8. Find an orthogonal matrix P and a diagonal matrix D such that pTAP-Duse the square root symbol 'where needed to give an exact value for your answer. -1 47 A- 4 2-4 0 0 0] P=10 0 0| D=10 0 0
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...
Let the matrix below act on C? Find the eigenvalues and a basis for each eigenspace...
Let the matrix below act on C? Find the eigenvalues and a basis for each eigenspace in c? 1 2 - 2 1 1 2 The eigenvalues of - 2 1 (Type an exact answer, using radicals and i as needed. Use a comma to separate answers as needed.) are A basis for the eigenspace corresponding to the eigenvalue a + bi, where b>0, is (Type an exact answer, using radicals and i as needed.) A basis for the eigenspace...
Let A be the matrix below. Find the singular values of A and enter them as...
Let A be the matrix below. Find the singular values of A and enter them as a comma-separated list. Use these to find a singular value decomposition A= ULV. Use the square root symbol 'V' where needed to give an exact value for your answer. 10 A= -5 -2 [01] Singular values: ??? To 0 0 0 0 0 T0 0 0] A = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...
1. The symmetric matrix [4 1 1 1 A = 4-1 1 -1 4 -1 4 -1 has eigenvalues A = 1 (with algebraic multiplicity 1) and A...
1. The symmetric matrix [4 1 1 1 A = 4-1 1 -1 4 -1 4 -1 has eigenvalues A = 1 (with algebraic multiplicity 1) and A 5 (with algebraic multiplicity 3). a) Find bases for the eigenspaces E(1) and E(5). b) Apply the Gram-Schmidt process to your basis for E(5) to find an onal basis for E (5) orthog- (c) Hence write down an that QT AQ = D. orthogonal matrix Q and a diagonal matrix D such...
Find the eigenvalues of the given matrix. [-14 -6 36 16 1) A) -2.-4 B)-4 C)-2...
Find the eigenvalues of the given matrix. [-14 -6 36 16 1) A) -2.-4 B)-4 C)-2 D) -24 The characteristic polynomial of a 5 5 matrix is given below. Find the eigenvalues and their multiplicities 2) A5 - 24A4-189A3-486A2 2) A) 0 (multiplicity 2),-9 (multiplicity 2),-6 (multiplicity 1) B) 0 (multiplicity 1),9 (multiplicity 3), 6 (multiplicity ) C) 0 (multiplicity 2),9 (multiplicity 2),6 (multiplicity 1) D) 0 (multiplicity 2),-9 (multiplicity 2),6 (multiplicity 1) Diagonalize A- PDP-1 the matrix A, if...
Let the matrix below act on C. Find the eigenvalues and a basis for each eigenspace...
Let the matrix below act on C. Find the eigenvalues and a basis for each eigenspace in c The eigenvalues of - 3 2 - 0 (Type an exact answer, using radicals and i as needed. Use a comma to separate answers as needed.)
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3,...
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
# 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...
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...
The symmetric matrix A below has distinct eigenvalues 10,-2 and-8. Find an orthogonal matrix P and a diagonal matrix D such that pTAP-Duse the square root symbol 'where needed to give an exact value for your answer. -1 47 A- 4 2-4 0 0 0] P=10 0 0| D=10 0 0
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...
Let the matrix below act on C? Find the eigenvalues and a basis for each eigenspace in c? 1 2 - 2 1 1 2 The eigenvalues of - 2 1 (Type an exact answer, using radicals and i as needed. Use a comma to separate answers as needed.) are A basis for the eigenspace corresponding to the eigenvalue a + bi, where b>0, is (Type an exact answer, using radicals and i as needed.) A basis for the eigenspace...
Let A be the matrix below. Find the singular values of A and enter them as a comma-separated list. Use these to find a singular value decomposition A= ULV. Use the square root symbol 'V' where needed to give an exact value for your answer. 10 A= -5 -2 [01] Singular values: ??? To 0 0 0 0 0 T0 0 0] A = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...
1. The symmetric matrix [4 1 1 1 A = 4-1 1 -1 4 -1 4 -1 has eigenvalues A = 1 (with algebraic multiplicity 1) and A 5 (with algebraic multiplicity 3). a) Find bases for the eigenspaces E(1) and E(5). b) Apply the Gram-Schmidt process to your basis for E(5) to find an onal basis for E (5) orthog- (c) Hence write down an that QT AQ = D. orthogonal matrix Q and a diagonal matrix D such...
Find the eigenvalues of the given matrix. [-14 -6 36 16 1) A) -2.-4 B)-4 C)-2 D) -24 The characteristic polynomial of a 5 5 matrix is given below. Find the eigenvalues and their multiplicities 2) A5 - 24A4-189A3-486A2 2) A) 0 (multiplicity 2),-9 (multiplicity 2),-6 (multiplicity 1) B) 0 (multiplicity 1),9 (multiplicity 3), 6 (multiplicity ) C) 0 (multiplicity 2),9 (multiplicity 2),6 (multiplicity 1) D) 0 (multiplicity 2),-9 (multiplicity 2),6 (multiplicity 1) Diagonalize A- PDP-1 the matrix A, if...
Let the matrix below act on C. Find the eigenvalues and a basis for each eigenspace in c The eigenvalues of - 3 2 - 0 (Type an exact answer, using radicals and i as needed. Use a comma to separate answers as needed.)
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
# 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...
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...
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