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- x'(t) = of the any - 2 5-6 4 1-4-6|x (t) x (0) = -2 + -3 1 The only eigen value of this matrix is -3, a triple root. You must explicitly find matrix involved, with the exception cannot involve Also, your answer 0 imaginary any matrix inverses. number i.Consider the Initial Value Problem: X1 = (a) Find the eigenvalues and eigenvectors for the coefficient matrix. 181 v1 x₁ x^2 = = - 3x₁ + 3x₂ 9 - 6x1 + 3x₂ " (b) Solve the initial value problem. Give your solution in real form. X1 = x2 = and A2 x₁ (0) x₂ (0) = " = = V2 = 2 7 [8]find the eigenvalues 3*3 matrix (-1 4 -2 -3 4 0 -3 1 3)
- X1(t) Let *(1) = be the solution to the discrete dynamical system: x2(t) X1(1 + 1) = 11x1(t) – 18x2(t) x2(t + 1) = 12x1(t) – 19x2(t) -1 If x(0) = find x(t). %3D -4 Put the eigenvalues in ascending order when you enter x1(t), x2(t) below. X1(t) = )'+ x2(t) = )'+ IIS + 3y - 2r =0 (b) Express the as a first order system, and then find the eigen- values.Use the power method to calculate an approximation to the dominant eigenpair for 2 4-(37) with X.-(1) A = -6 AºX₁.A³X₁ for 2. A³X₁.A³X₁ Estimate the error by using-
- Find the Eigenvalue(s) of the matrix Row 1: (-2,0,0,0) Row 2: (35,6,0,0) Row 3: (?,-12,4,0) Row 4: (8,?,0,-sqrt(2)) ?=number not givenA₁ = 5 + 2i with ₁ and 2 = 5-24 with ₂ = [21] Write the general real solution for the linear system' = A. In eigenvalue/eigenvector form: [20] ([ x(t) [2] -2i = = + C₂ B. In fundamental matrix form: 1 2 2 e^(5t)cos(2t) 2e^(5t)sin(2t) cos(2t) = C. As two equations: (write "c1" and "c2" for C₁ and C₂) x(t) c1e^(5t)cos(2t)+c2e^(5t)sin(2t) y(t) = c1*2e^(5t)sin(2t)-c2*2e^(5t)cos(2t) cos(2t) AT, in the following forms: + 0 2 19 2 e^(5t)sin(2t) -2e^(5t)cos(2t) sin(2t) sin(2t) Note: if you are feeling adventurous you could use other eigenvectors like 47₁ or -37₂. e 5 e 01 5 tLet A= (in the picture) (a) Find the eigenvalues and eigenvectors of A. (b) Rewrite A in the form A = P DP ^−1, where A, P and P −1 have their usual meanings and use this to find A^20. (c) Find a matrix B such that B^3 = A (Use A = PDP^−1).
- Consider the following matrix ^- [61] A = (a) Find the eigenvalues and the eigenvectors of this matrix. (You need to show how you find them.) (b) Using the eigenvalues and the eigenvectors of A, solve the system of differential equations X = AX.2) Find the eigenvalues and associated eigenfunctions for the equation Y" + AY = 0, ^ > 0 and Y(0) = 0, Y(8) = 0.Find all solution to the eigenvalue problem -X" = A X. X (-L) = X (L) & X'(-L) = X'(L)
