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Consider the 'Gibonacci' numbers G₁ defined through the recurrence relation G₁+2 G+1+Gk, with Go=0 and G₁-1. This two-term recurrence relation can be converted into a one-term recurrence relation as follows: G₁+2=G₁+1+G₂ Gk+1=Gk+1 and if we define the column vector (G+1, G₁)¹, we have U+1 - Aug where A is a 2 x 2 matrix. i) Find the matrix A, and show that u = A*u. What is u? ii) Find the eigenvalues of A, and show that its eigenvectors are of the form x'- (A,, 1). You need not normalise these eigenvectors.

Consider the 'Gibonacci' numbers G₁ defined through the recurrence relation G₁+2 G+1+Gk, with Go=0 and G₁-1. This two-term recurrence relation can be converted into a one-term recurrence relation as follows: G₁+2=G₁+1+G₂ Gk+1=Gk+1 and if we define the column vector (G+1, G₁)¹, we have U+1 - Aug where A is a 2 x 2 matrix. i) Find the matrix A, and show that u = A*u. What is u? ii) Find the eigenvalues of A, and show that its eigenvectors are of the form x'- (A,, 1). You need not normalise these eigenvectors.

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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Consider the 'Gibonacci' numbers Gg defined through the recurrence relation G+2 G+1 + Gk, with Go = 0 and G1 = 1. This two-term recurrence relation can be converted into a one-term recurrence relation as follows: G+2 Ge+1 = Gr+1; %3D and if we define the column vector u, = (Gr+, Ga)", we have u+1 = Aug, where A is a 2 x 2 matrix. i) Find the matrix A, and show that u, = A*u. What is u,? ii) Find the eigenvalues of A, and show that its eigenvectors are of the form x' = (A, 1)". You need not normalise these eigenvectors. iii) Using a result from an earlier part of this question, show that the Gibonacci number Gr approaches as k → o.
Transcribed Image Text:Consider the 'Gibonacci' numbers Gg defined through the recurrence relation G+2 G+1 + Gk, with Go = 0 and G1 = 1. This two-term recurrence relation can be converted into a one-term recurrence relation as follows: G+2 Ge+1 = Gr+1; %3D and if we define the column vector u, = (Gr+, Ga)", we have u+1 = Aug, where A is a 2 x 2 matrix. i) Find the matrix A, and show that u, = A*u. What is u,? ii) Find the eigenvalues of A, and show that its eigenvectors are of the form x' = (A, 1)". You need not normalise these eigenvectors. iii) Using a result from an earlier part of this question, show that the Gibonacci number Gr approaches as k → o.
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