Let A be the matrix, A =[2 4(a) Show that [ 1 -1 ] is an eigenvector of A and find the correspondingeigenvalue.(b) Find the other eigenvalue and a corresponding eigenvector.(c) Using the results in a) and b), or otherwise, solve the vector-matrixdifferential equationgiven that x(0) = [ 1 1]⁰.x- [14]xX,2

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Answered: Let A be the matrix, A = 2 (a) Show…

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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A 3×3 real matrix has three eigenvalues. One of them is λ₁ = Another is A₂ = 1 + i with eigenvector v2 - (3) = -1 with eigenvector v₁ = (a) What is the third eigenvalue and its corresponding eigenvector? (b) Find the general solution to x' = Ax, in purely real form. (c) Find the solution to x' = Ax subject to initial condition x (0) = (:). (6) 1
A 2x2 upper triangular matrix A has eigenvalues a and - 5 (with a ≠ -5). An eigenvector corresponding to the eigenvalue a is (1,0). Which of the following could be the matrix A? A. (-5 a/2 0) B. (a 0/2 -5) C. (-5 2/0 a) D. (a -5/0 2) E. (a 2/0 -5)
Suppose A is a 3 x 3 matrix with eigenvectors [1] V1 = V2 = 1 and v3 = corresponding to eigenvalues A1 =-, d2 = }, and A3 = 1, respectively. Find A and A20.
x1 = x2 – 4x + 4x3 %3D 4 x½ = -3x2 - V3x, + V3x1 +X3 x3 = 7x2 - 2x1 Find the eigenvalues and eigenvectors by using the above equations.
Given that the matrix A has eigenvaluesX₁ = -5 with corresponding eigenvector ₁ = and A₂ = 3 with corresponding eigenvector 2 = A = 1 [:] , find A. [:]
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]
For (a) and (b), 2 -1 [A] = -1 2 (a) What are the eigenvalues of A? (b) For each eigenvalue, find a nonzero eigenvector. [1/3] [1/4] (c) Suppose that the matrix B has the eigenvalues 1 and -6, with eigenvectors and respectively. Calculate Q(t).N(0)'. (Q: fundamental matrix for u’=B.u) (d) What is the solution to u'=B.u with u(0)=| ?
For the system of differential equations, X1, X2 a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₁ = U2 = = y' = c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u2 is the eigenvector associated with the larger eigenvalue A2. 5 2 - 1 Y2 (t) d) Determine two linearly independent solutions to the system. Enter the first solution in the format y₁ (t) y = = y₁ (t) Enter the second solution in the format y₂(t) = ƒ2(t) (v3h₁(t) + v₁h₂(t)) = : fi(t) (vigi(t) – v2g2(t)) . +
- (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
Q3. (a) If Ax = 2 x, determine the eigenvalues and the corresponding eigenvectors for A= G ) 2 3. Hence, write down the associated modal matrix P and diagonal matrix D, and use these values to solve the following system differential equations: *1 = X1 + 4 x2 X2 = 2 x1 + 3 x2 Given that when t = 0, x1 = 0 and x2 = 2. (b) Use the 4th order Runge Kutta method to solve the differential equation: dy e* y dx for values of x 0 (0.2) 0.4 given that y 1 when x 0. %3D Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x = 0.4 with the values obtained using Runge Kutta,
For the following matrix: a) find the eigenvalues and the eigenvectors b) check the answer by using the orthogonal property (x(i))T(x(j))=1(i≠l) c) determine the three normalized eigenvectors by using (x(i))T(x(i))=1(i≠l)
Let M be a 2 x 2 matrix with eigenvalues A₁ = 0.3, A2 = -0.25 with corresponding eigenvectors = [2²] ²2² = [9] · V2 Consider the difference equation with initial condition xo That is, write xo = C1V1 + C₂V2 H Write the initial condition as a linear combination of the eigenvectors of M. In general, X = Specifically, X4 V1 = For large k, xk Xk+1 = V₁+ )k v₁+ Mxk A V2 7)k V₂

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