The Fibonacci sequence (Fn)n=1,2,... is defined recursively as 1, n = 1, 1, n = 2, Fn-1 + Fn-2, n > 2. Fn = Let n = [Fn, Fn-1], for n = 2, 3, ... (a) Find a matrix A € R2x2 such that Axn-1, (b) Let X € R²x2 and the diagonal matrix A € R²×2 be such that A = XAX-¹. Show that x₂ = XA¹-²X-¹x₂, n> 2. Compute X and A (for example in Matlab using [X,L] = eig(A)). Use the decom- position to compute also F10, F20 and F30- Xn = n > 2.
The Fibonacci sequence (Fn)n=1,2,... is defined recursively as 1, n = 1, 1, n = 2, Fn-1 + Fn-2, n > 2. Fn = Let n = [Fn, Fn-1], for n = 2, 3, ... (a) Find a matrix A € R2x2 such that Axn-1, (b) Let X € R²x2 and the diagonal matrix A € R²×2 be such that A = XAX-¹. Show that x₂ = XA¹-²X-¹x₂, n> 2. Compute X and A (for example in Matlab using [X,L] = eig(A)). Use the decom- position to compute also F10, F20 and F30- Xn = n > 2.
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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![4. The Fibonacci sequence (Fn)n=1,2,... is defined recursively as
1,
n = 1,
1,
n = 2,
Fn-1 + Fn-2, n > 2.
Fn
=
Let n = [Fn, Fn-1], for n = 2, 3, ...
(a) Find a matrix A € R2×2 such that
Axn-1,
(b) Let X € R²×2 and the diagonal matrix A € R²×² be such that A XAX-¹. Show that
x₁ = XA¹-² X-¹x₂,
n > 2.
Compute X and A (for example in Matlab using [X,L] = eig(A)). Use the decom-
position to compute also F10, F20 and F30-
Xn
n > 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F456e193f-2af7-4cbe-b4f0-ceb3dbbc38b3%2Fbd5509b4-2823-4203-ac59-13d1afd4e624%2Fztqt4q_processed.png&w=3840&q=75)
Transcribed Image Text:4. The Fibonacci sequence (Fn)n=1,2,... is defined recursively as
1,
n = 1,
1,
n = 2,
Fn-1 + Fn-2, n > 2.
Fn
=
Let n = [Fn, Fn-1], for n = 2, 3, ...
(a) Find a matrix A € R2×2 such that
Axn-1,
(b) Let X € R²×2 and the diagonal matrix A € R²×² be such that A XAX-¹. Show that
x₁ = XA¹-² X-¹x₂,
n > 2.
Compute X and A (for example in Matlab using [X,L] = eig(A)). Use the decom-
position to compute also F10, F20 and F30-
Xn
n > 2.
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