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- Let 1₂ Calculate the eigenvalues of A and an eigenvector corresponding to the greatest eigenvalue. X₁ X3 = || = || A = X = 0 1 0 0 0 1 18 15 –41-3 2) 2) Verify that F = 0 -4 3 has eigenvalue/eigenvector pairs 2, 0-6 5) and compute D = Q¯'FQ. (1 2 1 20.₁.0-¹.0 () Write Q=1 0 2 20 23 *. Sketch a graph of the eigenvectors and plot Solve the system of ODES given by i' = | , some sample trajectories. Is the origin an attractor, a repeller or a saddle point? -2.
- Let A = Find the Characteristic Polynomial, and compute the Eigenvalues: They are (note i = v-1) is the iaginary "unit"): O p(A) = X² + 21 +1=A1 = -1, and A2 = -1, O p(A) = X² – 1=\1 = -1, and A2 = 1 O p(A) = X² – A +1=A1 = 1, and dz O p(A) = X² + 2A + 2 = A1 = -1+ i, and X2 = -1 - i O p(A) = X – 2A + 2 A1 = 1 + i, and A, = 1 – i O p(A) = X2 +1=\1 = i, and A2 = -i 1 W 000 000 F4 F3 E5Let x(t) x₁ (t) x' (t) x₁ (t): = = If x (0) = x₂(t): = = = ] Put the eigenvalues in ascending order when you enter x₁ (t), x₂(t) below. exp( t) + exp( x₁(t) x₂(t) -27 x₁(t) + 12x₂(t) -56 x₁(t) + 25 x₂(t) 4 be a solution to the system of differential equations: -2 exp( find x(t). t) + exp( t) t)(ii) Let A c R" be symmetric and 2 an eigenvalue of 4. J4| is a singular value of 4. (2)
- 2: Let 10 -1 -1 A = 0. 10 10 -1 -1 10 (a) Find the characteristic polynomial of A. (b) Find the dimension of the eigenspace corresponding to the largest eigenvalue.The matrix has two distinct eigenvalues with ₁Let A (a) Determine the eigenvalues X₁ and ₂ of A where X₁Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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