%3D Now consider the sequence defined by M1 = 0, M2 Mn = 2Mn-1 +3Mn-2, for n > 3. (a) Write down the first five terms of the sequence. (b) Write the matrix A satisfying [Ma-2] = A Mn-1 [Mn-1 for n > 3. Mn (c) Diagonalize A, that is, write A = PDP-1 for some diagonal matrix D.
%3D Now consider the sequence defined by M1 = 0, M2 Mn = 2Mn-1 +3Mn-2, for n > 3. (a) Write down the first five terms of the sequence. (b) Write the matrix A satisfying [Ma-2] = A Mn-1 [Mn-1 for n > 3. Mn (c) Diagonalize A, that is, write A = PDP-1 for some diagonal matrix D.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. Linear recurrence relation
(You will likely see this topic in a course on discrete math.)
The Fibonacci sequence, named after the Italian mathematician Leonard Fibonacci of Pisa,
is the most well-known sequence of numbers. A lot of the numbers in this sequence appear
in nature (see the video Nature by Numbers: The Golden Ratio and Fibonacci Numbers)
The first two terms of the Fibonacci sequence are
fi = 1
and f2 = 1.
The subsequent terms of the Fibonacci sequence are given by the recurrence relation
fn
fn-1+ fn-2,
for n > 3.
||
Here are the first few terms of the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
Let A =
Observe that
A
A?
= A
and A3
In general, we have
|fn-1
[fn-2
[fn-1]
= A
and
= An-2
for n > 3.
fn
fn-1]
fn
1
This matrix A has two eigenvalues 1tv5 and y with eigenvectors
1
and
-V5 :
1+/5
2
respectively. The diagonalization of A is PDP-l where
1
1 -1+V5
1
1+v5
P =
P-1
and
D =
2
1+/5
1-V5 )
V5
1+V5
1-V5
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51be4038-25b0-462c-88b7-f9d17375dd0c%2Fc2ce6c49-a05a-4207-a448-21bbea8c890a%2Fye4e0t_processed.png&w=3840&q=75)
Transcribed Image Text:1. Linear recurrence relation
(You will likely see this topic in a course on discrete math.)
The Fibonacci sequence, named after the Italian mathematician Leonard Fibonacci of Pisa,
is the most well-known sequence of numbers. A lot of the numbers in this sequence appear
in nature (see the video Nature by Numbers: The Golden Ratio and Fibonacci Numbers)
The first two terms of the Fibonacci sequence are
fi = 1
and f2 = 1.
The subsequent terms of the Fibonacci sequence are given by the recurrence relation
fn
fn-1+ fn-2,
for n > 3.
||
Here are the first few terms of the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
Let A =
Observe that
A
A?
= A
and A3
In general, we have
|fn-1
[fn-2
[fn-1]
= A
and
= An-2
for n > 3.
fn
fn-1]
fn
1
This matrix A has two eigenvalues 1tv5 and y with eigenvectors
1
and
-V5 :
1+/5
2
respectively. The diagonalization of A is PDP-l where
1
1 -1+V5
1
1+v5
P =
P-1
and
D =
2
1+/5
1-V5 )
V5
1+V5
1-V5
2
Transcribed Image Text:As we will see in Section 7.3.1,
A? = (PDP-1)(PDP 1)= PD(P-1P)DP-1 = PDIDP-1 = PD²P¬1
and
Ak = PD*P-1
for k > 1.
As a result, we get the numbers in the Fibonacci sequence as follows:
fn-1
An-2
fn
PDn-2p-1
п-2
1
1 =1+v5
1
1
2
-
1+/5
1-V5
()
n-2
V5
1+v5
n-1
п-1-
1+/5
/5
Võ[ ()" - (4)
We conclude that, for n > 1, the n-th term of the Fibonacci sequence is
n
1+V5
1
fn
V5
1-
2
Note that y5 is called the golden ratio.
Now consider the sequence defined by M1 = 0, M2 = 1 and
Мn 3 2Mп-1 + 3Mп-2,
for n > 3.
(a) Write down the first five terms of the sequence.
(b) Write the matrix A satisfying
|Mn-1
|Mn-2
= A
Mn
for n > 3.
Mn-1
(c) Diagonalize A, that is, write A = PDP-1 for some diagonal matrix D.
(d) Use the fact that Ak = PD*P-lto get a formula for Mn.
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