7. (a) Consider the following system of linear difference equations: Xn+1 -2xn - 2²n Yn+1 = xn + 3yn Zn Zn+1 = 2n + 3%n with coyo = 20 = 1. = (i) Write this system in the form Vn+1 = Avn, where A is the 3×3 coefficient matrix of the system and V₁ = [2]. Yn (ii) Calculate the eigenvalues and eigenvectors of the coefficient matrix A. (iii) Diagonalise the coefficient matrix A. (iv) Use your answer to part (iii) to find a closed form solution to the system of difference equations.
7. (a) Consider the following system of linear difference equations: Xn+1 -2xn - 2²n Yn+1 = xn + 3yn Zn Zn+1 = 2n + 3%n with coyo = 20 = 1. = (i) Write this system in the form Vn+1 = Avn, where A is the 3×3 coefficient matrix of the system and V₁ = [2]. Yn (ii) Calculate the eigenvalues and eigenvectors of the coefficient matrix A. (iii) Diagonalise the coefficient matrix A. (iv) Use your answer to part (iii) to find a closed form solution to the system of difference equations.
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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![7. (a) Consider the following system of linear difference equations:
Xn+1 -2xn - 2²n
Yn+1 = xn + 3yn Zn
Zn+1 = 2n + 3%n
with coyo = 20 = 1.
=
(i)
Write this system in the form Vn+1 = Avn, where A is the 3×3 coefficient matrix
of the system and V₁ = []).
Yn
(ii)
Calculate the eigenvalues and eigenvectors of the coefficient matrix A.
(iii) Diagonalise the coefficient matrix A.
(iv)
Use your answer to part (iii) to find a closed form solution to the system of
difference equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd63740e0-8dd1-40ff-ae15-48815d78b54a%2F7d91a01f-ac10-43b2-ba7d-c44fb5ce6290%2Fklhldl_processed.png&w=3840&q=75)
Transcribed Image Text:7. (a) Consider the following system of linear difference equations:
Xn+1 -2xn - 2²n
Yn+1 = xn + 3yn Zn
Zn+1 = 2n + 3%n
with coyo = 20 = 1.
=
(i)
Write this system in the form Vn+1 = Avn, where A is the 3×3 coefficient matrix
of the system and V₁ = []).
Yn
(ii)
Calculate the eigenvalues and eigenvectors of the coefficient matrix A.
(iii) Diagonalise the coefficient matrix A.
(iv)
Use your answer to part (iii) to find a closed form solution to the system of
difference equations.
Transcribed Image Text:(b)Let M be an nxn matrix.
(i)
Suppose u and v are eigenvectors of M with eigenvalue k. Show that any linear
combination of u and v is also an eigenvector of M with eigenvalue k.
(ii)
If w is an eigenvector of M with eigenvalue h, show that w is also an eigenvector
of Mm with eigenvalue hm, for any m € N.
(iii)
If k₁ and k2 are distinct eigenvalues of M, show that their corresponding
eigenvectors are linearly independent.
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