Consider the system of linear differential equations:X = AX.(i) Suppose is an eigen value of A of multiplicity k > 1. Define a root vector corresponding tothe eigen value \.(ii) Show that each eigen value \ of A of multiplicity k has k linearly independent root vectors.(iii) Suppose that A is an eigen value of A of multiplicity k and Uλ,1, Ux‚2, ……., Ux,k are root vectorscorresponding to λ of orders 1, 2, k, respectively. Further, let:...92!Xi(t) = e¼ [Ux,i + tŪλ,i−1 + U\‚i−2 + …...Show that X1(t), X2(t), . . ., Xk(t){X₁(t), X2(t),...,x(t)}ti-1(i − 1)!-¡Uλ,1], for i = 1, 2, … . . k.is a set of linearly independent solutions of the systemof linear differential equations and every solution of the system is a linear combination of thesolutions in this set.

A generalized eigenvector of a matrix A corresponding to an eigenvalue λ is a nonzero vector x that…

Answered: Consider the system of linear…

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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For an electrical circuit containing an inductor, a resistor, and a capacitor, the charge Q(t) on the capacitor satisfies the ODE 8²³Q +2 dt² dQ dt +2 +2Q=0, (in suitably scaled variables). Write the equation as a system of first-order ODEs and find the general solution using the eigenvector method.
(3) Let matrix A have eigenvectors . . associated with eigenvalues Xo = 4, A₁ = 2, A₂ = 0), respectively. Suppose some vector 7 is: 7 = №n +27₁ +30¹₂ Compute Ar. Give your result as a linear combination of eigenvectors ₁.
2. (1) Compute all eigenvalues and eigenvectors of the Cartan matrix 2 -1 C = -1 2 -1 -1 2 (2) Diagonalize C (i.e. find Q such that Q-'CQ is diagonal). (3) Is C positive definite?
Verify that 1, is an eigenvalue of A and that x, is a corresponding eigenvector. = 13, x1 12 = -3, x, = (-2, 1 0) 13 = -3, x3 = (3, 0, 1) -1 4 -6 (1, 2, –1) A = 4 5 -12 -2 -4 -1 4 -6 1 1 = 13 2 = 1,x1 4 5 -12 = -2 -4 3 -1 -1 -1 4 -6 -2 -2 AX2 1 = 12x2 4 5 -12 1 = -2 -4 -1 4 -6 3 3 AX3 = 13x3 4 5 -12 -3 = -2 -4 1
A matrix A E M(3, R) that has as eigenvalues 1, -1, -1 associated respectively to the eigenvectors (50, 50, -50), (0,16,16) and (24,0, -24) corresponds to?
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)) . +
b) Consider the matrix TM UTM UTM UTM -1 21 A = OTM UTM 7TMUTM UTM 4 i. Compute the dominant eigenvalue, di and the corresponding eigen- -4 20 UT vector x1 for the matrix A using power method starting with x0) = (1, 1, 1)" and stop the iterations when |mg+1 – mel < 0.05. AUTM %3D ii. Determine the smallest eigenvalue, Az for matrix A by using shifted power method based on the results in (i). Start the iteration with initial vector x) = (1,0,0)" and stop the iterations when ||x*+1) – x*)|| < 0.05. TM UTM UTM (k) %3D
- (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
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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