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Question 5 [3+(2+4) marks] (a) The matrix A has a repeated eigenvalue of 1 = 2....
(1 point) Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized...
(1 point) Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized eigenvector: i = -3 with eigenvector v = and generalized eigenvector w= = [-] = [4] Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: 1 (O) = - 18.05.8:8)... y(t) B. In fundamental matrix form: (O)- x(1) y(t) [:] C. As two equations: (write "c1" and "c2" for c and c2) X(t) = yt)...
(1 point) Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized...
(1 point) Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized eigenvector: X= -4 with eigenvector v = and generalized eigenvector ū= [] (-1) Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: t t [CO] = C1 + C2 + I g(t). e . - 1 B. In fundamental matrix form: [CO] C. As two equations: (write "c1" and "c2" for 1 and 2) X(t)...
A) B) (1 point) The matrix A= 1-3 0 [1 0 -4 0 -1] 0 -5...
A)
B)
(1 point) The matrix A= 1-3 0 [1 0 -4 0 -1] 0 -5 has one real eigenvalue. Find this eigenvalue and a basis of the eigenspace. The eigenvalue is -4 A basis for the eigenspace is (1 point) Find the solution to the linear system of differential equations x' y' = = 25x + 727 9 -9.2 – 26y satisfying the initial conditions x(0) = -18 and y(0) = 7. x(t) = y(t) =
4. Let E) 6 3 0 [8 Marks] 3 6 0 A = 0 0 11...
4. Let E) 6 3 0 [8 Marks] 3 6 0 A = 0 0 11 a) Find the eigenvalues of A b) For each eigenvalue of A, find a basis for the corresponding eigenspace. c) Consider the linear transformation T : R3 -> R3 defined by T(x) = Ax for every xE R3. Find a basis of R3 in which the matrix representing T is diagonal
3 0 2 0 The matrix A=11 3 1 0 10 has eigenvalue t. Find a basis for the eigenspace E9) 0 0 0 4
3 0 2 0 The matrix A=11 3 1 0 10 has eigenvalue t. Find a basis for the eigenspace E9) 0 0 0 4
3 0 2 0 The matrix A=11 3 1 0 10 has eigenvalue t. Find a basis for the eigenspace E9) 0 0 0 4
2 (5 points) Recalled that null space of a matrix A € Mnxn is defined as...
2 (5 points) Recalled that null space of a matrix A € Mnxn is defined as N(A) = {r € R” : Ar =0}. Now, the eigenspace of A corresponding to the eigenvalue 1 (denoted by Ex(A)) is defined as the nullspace of A-XI, that is, EX(A) = N(A – XI) = {v ER”: (A – XI)v = 0}. You should have three distinct eigenvalues in Problem 1 above. Let say there are li, 12, and 13. (i) Find the...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
A9.5.36 Question Help Find a general solution to the system below. -2 x(t) x'(t) = This...
A9.5.36 Question Help Find a general solution to the system below. -2 x(t) x'(t) = This system has a repeated eigenvalue and one linearly independent eigenvector. To find a general solution, first obtain a nontrivial solution x, (t). Then, to obtain a second linearly independent solution, try x2 (t) = te"u, + e"u2, where r is the eigenvalue of the matrix and u, is a corresponding eigenvector. Use the equation (A - rl)u, = u, to find the vector u,....
Ui l uentical . i Let A be a square matrix of order n and λ be an eigenvalue of A with geometric ...
ui l uentical . i Let A be a square matrix of order n and λ be an eigenvalue of A with geometric multiplicity k, where 1kn. Choose a basis B -(V1, v2,. .. , Vk) of &A) and extend this to a basis B of R". (1) Show that the matrix of the linear transformation x Ax on R" induced by the matrix A with respect the basis B on both the domain and codomain is:
ui l uentical...
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
(1 point) Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized eigenvector: i = -3 with eigenvector v = and generalized eigenvector w= = [-] = [4] Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: 1 (O) = - 18.05.8:8)... y(t) B. In fundamental matrix form: (O)- x(1) y(t) [:] C. As two equations: (write "c1" and "c2" for c and c2) X(t) = yt)...
(1 point) Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized eigenvector: X= -4 with eigenvector v = and generalized eigenvector ū= [] (-1) Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: t t [CO] = C1 + C2 + I g(t). e . - 1 B. In fundamental matrix form: [CO] C. As two equations: (write "c1" and "c2" for 1 and 2) X(t)...
A)
B)
(1 point) The matrix A= 1-3 0 [1 0 -4 0 -1] 0 -5 has one real eigenvalue. Find this eigenvalue and a basis of the eigenspace. The eigenvalue is -4 A basis for the eigenspace is (1 point) Find the solution to the linear system of differential equations x' y' = = 25x + 727 9 -9.2 – 26y satisfying the initial conditions x(0) = -18 and y(0) = 7. x(t) = y(t) =
4. Let E) 6 3 0 [8 Marks] 3 6 0 A = 0 0 11 a) Find the eigenvalues of A b) For each eigenvalue of A, find a basis for the corresponding eigenspace. c) Consider the linear transformation T : R3 -> R3 defined by T(x) = Ax for every xE R3. Find a basis of R3 in which the matrix representing T is diagonal
3 0 2 0 The matrix A=11 3 1 0 10 has eigenvalue t. Find a basis for the eigenspace E9) 0 0 0 4
3 0 2 0 The matrix A=11 3 1 0 10 has eigenvalue t. Find a basis for the eigenspace E9) 0 0 0 4
2 (5 points) Recalled that null space of a matrix A € Mnxn is defined as N(A) = {r € R” : Ar =0}. Now, the eigenspace of A corresponding to the eigenvalue 1 (denoted by Ex(A)) is defined as the nullspace of A-XI, that is, EX(A) = N(A – XI) = {v ER”: (A – XI)v = 0}. You should have three distinct eigenvalues in Problem 1 above. Let say there are li, 12, and 13. (i) Find the...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
A9.5.36 Question Help Find a general solution to the system below. -2 x(t) x'(t) = This system has a repeated eigenvalue and one linearly independent eigenvector. To find a general solution, first obtain a nontrivial solution x, (t). Then, to obtain a second linearly independent solution, try x2 (t) = te"u, + e"u2, where r is the eigenvalue of the matrix and u, is a corresponding eigenvector. Use the equation (A - rl)u, = u, to find the vector u,....
ui l uentical . i Let A be a square matrix of order n and λ be an eigenvalue of A with geometric multiplicity k, where 1kn. Choose a basis B -(V1, v2,. .. , Vk) of &A) and extend this to a basis B of R". (1) Show that the matrix of the linear transformation x Ax on R" induced by the matrix A with respect the basis B on both the domain and codomain is:
ui l uentical...
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
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