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Consider the ‘Gibonacci’ numbers Gk defined through the recurrence relation Gk+2 =1/2Gk+1 + 1/2Gk, with G0 = 0 and G1 = 1. This two-term recurrence relation can be converted into a one-term recurrence relation as provided (image), and if we define the column vector uk = (Gk+1, Gk)T, we have uk+1 = Auk, where A is a 2 × 2 matrix. Find the eigenvalues of A, and show that its eigenvectors are of the form xi = (λi, 1)T. You need not normalise these eigenvectors.

Consider the ‘Gibonacci’ numbers Gk defined through the recurrence relation Gk+2 =1/2Gk+1 + 1/2Gk, with G0 = 0 and G1 = 1. This two-term recurrence relation can be converted into a one-term recurrence relation as provided (image), and if we define the column vector uk = (Gk+1, Gk)T, we have uk+1 = Auk, where A is a 2 × 2 matrix. Find the eigenvalues of A, and show that its eigenvectors are of the form xi = (λi, 1)T. 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 Gk defined through the recurrence relation Gk+2 =1/2Gk+1 + 1/2Gk, with G0 = 0 and G1 = 1. This two-term recurrence relation can be converted into a one-term recurrence relation as provided (image), and if we define the column vector uk = (Gk+1, Gk)T, we have uk+1 = Auk, where A is a 2 × 2 matrix.

Find the eigenvalues of A, and show that its eigenvectors are of the form xi = (λi, 1)T. You need not normalise these eigenvectors. 

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