Suppose that the matrix A has the following eigenvalues and eigenvectors: 121 = 2i with vị =-4 + 5iand= -2i with v2=-4 – 5iWrite the general real solution for the linear system r' = Ar, in the following forms:A. In eigenvalue/eigenvector form:).x(t)t= C1e^(2i)e^(2i)y(t)-45ie^(-21) +e^(2i)e-45iB. In fundamental matrix form:cos(2t)sin(2t)x(t)y(t),-4cos(2t)-5sin(2t)-4cos(2t)+5sin(2t)C. As two equations: (write "c1" and "c2" for cq and c2 )x(t) == c1cos(2t)+c2sin(2t)y(t)-5c1sin(2t)-4c2cos(2t)

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Answered: 21 2i with vi = -4 + 5i and = -2i with…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.1: Eigenvalues And Eigenvectors

Problem 65E

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Consider the linear system X1 = and Find the eigenvalues and eigenvectors for the coefficient matrix. A₂ = 0 JI ← A, V₁ 9 V2 = 81 = = -3 6 - - 4 x. help (numbers) help (matrices) help (numbers) help (matrices) For each eigenpair in the previous part, form a solution of a' = Ax. Use t as the independent variable in your answers.
A₁ = 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 t
S + 3y - 2r =0 (b) Express the as a first order system, and then find the eigen- values.
What are the Eigenvalues of the system below: X' 1 -(-2²) 1 O 2+i, 2-i O 1,2 O 1+2i, 1-2i O +2i, -2i X
Let 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 linear system a. Find the eigenvalues and eigenvectors for the coefficient matrix. (-3+i) A₁ = 1 , vi b. Find the real-valued solution to the initial value problem [{ and A₂ = -3y1 - 2/2, 591 +33/2, Use t as the independent variable in your answers. vi(t) =(5-5/2i)e^(-it)(-3/5-1/5)+(5+5/2i)e^(it)(-3/5+i/5) 32(t)= (5-5/2)^(-it)+(5+5/2)^(it) 4 vo = 3/1 (0) = 6, 3/2 (0) = -5. (-3-1)/
Write the system of differential equations given in the 1st photo in the normal form defined in the 2nd photo and solve the resulting system by means of the eigenvalues and eigen-vectors of the square matrix A.
find the eigenvalues 3*3 matrix (-1 4 -2 -3 4 0 -3 1 3)
Pi- Determine which f th falining equations San an eigen funetion equation , and qive th valuet sinfIx) TTX a) 2. XP d sin(" dx2 iky
a) Find the eigenvalues of A. b) Find three linearly independent solutions of x' = Ax. c) Find the exponential e^(tA) Let A = 2 -1 ا دیا 0 0 3 0 -30
Suppose that the matrix A has the following eigenvalues and eigenvectors: A₁ = 4 + 3i with ₁ 54]. and A₂-432 with 7₂ 5i Write the general real solution for the linear system' = Ar, in the following forms: = A. In eigenvalue/eigenvector form: C-8--80). cos(t) [3] + C₂ y(t) = -4 + 5i 4 B. In fundamental matrix form: -4 4 -4 sin(t) C. As two equations: (write "c1" and "c2" for C₁ and C₂) x(t) = + 5 0 5 sin(t) cos(t) -4 e -4 t t
For the system of differential equations, = A₁, A₂ = U₁ = U2 = y' a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) = [16 b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. = -16 21 C1 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 X₂ . Enter the eigenvectors as a matrix with an appropriate size. 14 19 y d) Determine a general solution to the system. Enter your answer in the format y(t) = c₁f₁(t)ví + c₂f₂(t)v₂ . y(t) = + C₂

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