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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 = [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
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.
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.
(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.
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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