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3. +-/3 points Prove that if A2 o, then 0 is the only eigenvalue of A....
Q4. Let 1.01 0.99 0.99 0.98 (a) Find the eigenvalue decomposition of A. Recall that λ is an eigenvalue of A if for some u1],u2 (entries of the corresponding eigenvector) we have (1.01 u0.99u20 99...
Q4. Let 1.01 0.99 0.99 0.98 (a) Find the eigenvalue decomposition of A. Recall that λ is an eigenvalue of A if for some u1],u2 (entries of the corresponding eigenvector) we have (1.01 u0.99u20 99u [1] + (0.98-A)u[2] = 0. Another way of saying this is that we want the values of λ such that A-λ| (where I is the 2 x 2 identity matrix) has a non-trivial null space there is a nonzero vector u such that (A-AI)u =...
linear algebra (1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A,...
linear algebra
(1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A, then is an eigenvalue of A! Proof: Suppose v is an eigenvector of eigenvalue then Au=du. Since A is invertible, we can multiply both sides of Au= du by 50 Az = Azj. This implies that . Since 1 + 0 we obtain that Thus – is an eigenvalue of A-? A.D=AU B. A=X co=A D. X-A7 = E. A- F. Av= < P...
I need all details. Thx 2. Give an example of a matrix with the indicated properties. If the property cannot be attained, explain why not (a) A is 2 x 4 and has rank 3. (b) A is 3 × 3 and has dete...
I need all details. Thx
2. Give an example of a matrix with the indicated properties. If the property cannot be attained, explain why not (a) A is 2 x 4 and has rank 3. (b) A is 3 × 3 and has determinant 1. (c) A is 3 × 6 and has a 3 dimensional row space and a 6 dinensional column space (d) A is 3 × 3 and has a 2 dimensional null space. (e) A is...
(i) Prove that there are infinitely many elements in Eλ(A) for every eigenvalue λ of every 3 × 3 ...
(i) Prove that there are infinitely many elements in Eλ(A) for every eigenvalue λ of every 3 × 3 matrix A. (ii) Is the row space of (0 1 3 0 −1 4 0 2 −1) equal to R^3 ? Justify your answer.
(3 points) Given the system 1. -2 0 2i and for the eigenvalue λ-2, the vector V-(1) is an eigenvector. we know that...
(3 points) Given the system 1. -2 0 2i and for the eigenvalue λ-2, the vector V-(1) is an eigenvector. we know that λ- (a) find the general solution; (b) determine if the origin is a spiral sink, a spiral source, or a center; (e) determine the direction of the oscillation in the phase plane (do the solutions go clockwise or countercdlocdkwise around the origin?); or counterclockwise
(3 points) Given the system 1. -2 0 2i and for the eigenvalue...
(8 points) [102] The matrix A= 0 3 0 (205 has a single real eigenvalue =...
(8 points) [102] The matrix A= 0 3 0 (205 has a single real eigenvalue = 3 with algebraic multiplicity three (a) Find a basis for the associated eigenspace. Basis = { (b) is the matrix A defective? A. A is not defective because the eigenvectors are linearly independent O B. A is defective because the geometric multiplicity of the eigenvalue is less than the algebraic multiplicity c. A is defective because it has only one eigenvalue D. A is...
Question 4. The spectral decomposition (or the orthogonal eigenvalue decomposi- tion) of a matrix A whose...
Question 4. The spectral decomposition (or the orthogonal eigenvalue decomposi- tion) of a matrix A whose determinant is zero is given by A = (2) [11* • -*] +/- +] + (-1). tao ta + (e)- vv V2 for some v € Ry, and a real number c ER. (a) (5 points) Find the eigenvalues of A and the value of c. You must justify your answer. (b) (5 points) Find v. (c) (5 points) The matrix A can expressed...
True False a) For nxn A, A and AT can have different eigenvalues. b) The vector v 0 cannot be an eigenvector of A. c) If λ's an eigenvalue of A, then λ2 is an eigenvalue of A2. True False d) If A...
True False a) For nxn A, A and AT can have different eigenvalues. b) The vector v 0 cannot be an eigenvector of A. c) If λ's an eigenvalue of A, then λ2 is an eigenvalue of A2. True False d) If A is invertible, then A is diagonalizable. e) If nxn A is singular, then Null(A) is an eigenspace of A. f) For nxn A, the product of the eigenvalues is the trace of A. True False g) If...
1, and 6. An n xn matrix A is called idempotent if A2 = A. Some...
1, and 6. An n xn matrix A is called idempotent if A2 = A. Some examples include lude [22] fool the identity In: Idempotents correspond to "projections onto a subspace," as we will discuss later. Prove the following statements: a) If A is idempotent then so is A". b) If A is idempotent, then so is In - A. c) If A and B are both idempotent, and AB = BA= Onxn (the zero matrix), then A+B is idempotent....
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...
Q4. Let 1.01 0.99 0.99 0.98 (a) Find the eigenvalue decomposition of A. Recall that λ is an eigenvalue of A if for some u1],u2 (entries of the corresponding eigenvector) we have (1.01 u0.99u20 99u [1] + (0.98-A)u[2] = 0. Another way of saying this is that we want the values of λ such that A-λ| (where I is the 2 x 2 identity matrix) has a non-trivial null space there is a nonzero vector u such that (A-AI)u =...
linear algebra
(1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A, then is an eigenvalue of A! Proof: Suppose v is an eigenvector of eigenvalue then Au=du. Since A is invertible, we can multiply both sides of Au= du by 50 Az = Azj. This implies that . Since 1 + 0 we obtain that Thus – is an eigenvalue of A-? A.D=AU B. A=X co=A D. X-A7 = E. A- F. Av= < P...
I need all details. Thx
2. Give an example of a matrix with the indicated properties. If the property cannot be attained, explain why not (a) A is 2 x 4 and has rank 3. (b) A is 3 × 3 and has determinant 1. (c) A is 3 × 6 and has a 3 dimensional row space and a 6 dinensional column space (d) A is 3 × 3 and has a 2 dimensional null space. (e) A is...
(3 points) Given the system 1. -2 0 2i and for the eigenvalue λ-2, the vector V-(1) is an eigenvector. we know that λ- (a) find the general solution; (b) determine if the origin is a spiral sink, a spiral source, or a center; (e) determine the direction of the oscillation in the phase plane (do the solutions go clockwise or countercdlocdkwise around the origin?); or counterclockwise
(3 points) Given the system 1. -2 0 2i and for the eigenvalue...
(8 points) [102] The matrix A= 0 3 0 (205 has a single real eigenvalue = 3 with algebraic multiplicity three (a) Find a basis for the associated eigenspace. Basis = { (b) is the matrix A defective? A. A is not defective because the eigenvectors are linearly independent O B. A is defective because the geometric multiplicity of the eigenvalue is less than the algebraic multiplicity c. A is defective because it has only one eigenvalue D. A is...
Question 4. The spectral decomposition (or the orthogonal eigenvalue decomposi- tion) of a matrix A whose determinant is zero is given by A = (2) [11* • -*] +/- +] + (-1). tao ta + (e)- vv V2 for some v € Ry, and a real number c ER. (a) (5 points) Find the eigenvalues of A and the value of c. You must justify your answer. (b) (5 points) Find v. (c) (5 points) The matrix A can expressed...
True False a) For nxn A, A and AT can have different eigenvalues. b) The vector v 0 cannot be an eigenvector of A. c) If λ's an eigenvalue of A, then λ2 is an eigenvalue of A2. True False d) If A is invertible, then A is diagonalizable. e) If nxn A is singular, then Null(A) is an eigenspace of A. f) For nxn A, the product of the eigenvalues is the trace of A. True False g) If...
1, and 6. An n xn matrix A is called idempotent if A2 = A. Some examples include lude [22] fool the identity In: Idempotents correspond to "projections onto a subspace," as we will discuss later. Prove the following statements: a) If A is idempotent then so is A". b) If A is idempotent, then so is In - A. c) If A and B are both idempotent, and AB = BA= Onxn (the zero matrix), then A+B is idempotent....
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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