15. Find the eigenvalues and cigenvectors of matrix A. Then find the generalsolution to X(t + 1) = AX(t).(a) A=1 2()2()030-3(b) A =(43)

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Answered: 15. Find the eigenvalues and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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11) Find the values of such that the equations have non-trivial solutions. What are the eigenvalues and eigenvectors? (6-2) x + 3y = 0 3x - (2+2)y = 0
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-
(b) Determine the eigenvalues and eigenvectors of the matrix Hence, write down the associated modal matrix P and diagonal spectrum matrix D, and use these to solve the following system of differential equations * = 4x-2y j =x+y
Use (a) to solve the system x = (: :) . Х. -8 7 NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly independent solutions. The second will yield solutions which are identical (up to a constant) to the solutions already found. a) is as follows Show from first principles, i.e., by using the definition of linear independence, that if x+iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then the two real solutions Y(t) = et (u cos bt – w sin bt) and Z(t) = eat (u sin bt + w cos bt) are linearly independent solutions of X = AX.
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]
- (1-3). Determine the eigenvalue and corres- 3. a) Given the matrix A = ponding eigenvector and generalized eigenvector of A. b) Determine the general solution of the system of linear differential equations: x₁ = I₁=I₂ x₂ = x1₁+3x₂
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.
can someone explain this for me, please ?

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15. Find the eigenvalues and cigenvectors of matrix A. Then find the general solution to X(1 + 1) = AX(t). (a) A = ( 2 3 2 0 0 -3 (6) 4- (13) A = 4