1. A linear homogenous recurrence relation of degree 2 with constant coefficients is a recurrence relation of the form an = Aan-1+ Ban 2. , Vn 2 2, (1) where A and B are fixed real numbers. (a) Show that this relation can be rewritten in a matrix form as follows an-1 M (2) an-2 where M is a 2 x 2 matrix (that you need to determine). (b) Show by induction that an
1. A linear homogenous recurrence relation of degree 2 with constant coefficients is a recurrence relation of the form an = Aan-1+ Ban 2. , Vn 2 2, (1) where A and B are fixed real numbers. (a) Show that this relation can be rewritten in a matrix form as follows an-1 M (2) an-2 where M is a 2 x 2 matrix (that you need to determine). (b) Show by induction that an
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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Transcribed Image Text:1. A linear homogenous recurrence relation of degree 2 with constant coefficients is a recurrence
relation of the form
an = Adn-1+ Ban 2, Vn 2 2,
(1)
where A and B are fixed real numbers.
(a) Show that this relation can be rewritten in a matrix form as follows
()-
an-1
= M
(2)
where M is a 2 x 2 matrix (that you need to determine).
(b) Show by induction that
a1
= M"
(c) We turn now our attention to the Fibonacci sequence (a particular case of (1)) given by
( Fo = 0, Fı =1
Fn-1+ Fn-2
(3)
(d) Write the value of the matrix M for the recurrence relation (3) and establish that
(:)-
F, = (0 1) M" ()
(4)
(e) In order to determine a closed formula for F, we need to calculate the value of M". For
this purpose we should diagonalize the matrix M, i.e. we need to find two matrices P
and D such that
M = PDP',
(5)
where D is a diagonal matrix with its diagonal entries being the eigenvalues of M. Find
matrices P and D such that equation (5) is satisfied.
(f) Using equation (5), show that
M = PD"P-1
(g) Show that if the diagonal matrix D =
then D" =
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