Example 4. Determining Eigenvalues and Eigenvectors Consider the system of computer-dog growth equations from Section 2.5. C'=3C + D or c' = Ac, where A = D' = 2C + 2D In Section 2.5 the eigenvalues and eigenvectors were given without any explanation of how they were found. Let us calculate them now. By Theorem 2 the eigenvalues are the zeros of the characteristic poly- nomial det(A-AI): det(AAI) = 3-A 2 1 2 - A =(3A)(2A)-1-2 = =(65+2)- 2 = 4 5A + A² = (4-x)(1-x) So the zeros of det(AAI) = (4A)(1A) are 4 and 1. (19) To find an eigenvector u for the eigenvalue 4, we must solve the system Au=4u or, by matrix algebra, (A-41)u = 0, where 3-4 1 A-41- = = 2 2 - We find that 2n='n 0= n + 'n- 2u₁ - 24₂ = 0 The second equation here is just -2 times the first equation (so it is superfluous). Then u is an eigenvector if u₁ = u₂, or equivalently if u is a multiple of [1, 1]. It is left as an exercise for the reader to verify that v = [1, 2] is an eigenvector for λ = 1 by showing that this v is a solution to Av v or (A - I)v = 0. (i) [ ] (ii) [4] 23. (a) Compute the eigenvalues of each of the following matrices. 4 (iii) [ ] 園 3 0 2 1 (iv) (v) -1 -3 -2 2 3 (b) Determine an eigenvector associated with the largest eigenvalue, using the method in Example 4, for the matrices in part (a).
Example 4. Determining Eigenvalues and Eigenvectors Consider the system of computer-dog growth equations from Section 2.5. C'=3C + D or c' = Ac, where A = D' = 2C + 2D In Section 2.5 the eigenvalues and eigenvectors were given without any explanation of how they were found. Let us calculate them now. By Theorem 2 the eigenvalues are the zeros of the characteristic poly- nomial det(A-AI): det(AAI) = 3-A 2 1 2 - A =(3A)(2A)-1-2 = =(65+2)- 2 = 4 5A + A² = (4-x)(1-x) So the zeros of det(AAI) = (4A)(1A) are 4 and 1. (19) To find an eigenvector u for the eigenvalue 4, we must solve the system Au=4u or, by matrix algebra, (A-41)u = 0, where 3-4 1 A-41- = = 2 2 - We find that 2n='n 0= n + 'n- 2u₁ - 24₂ = 0 The second equation here is just -2 times the first equation (so it is superfluous). Then u is an eigenvector if u₁ = u₂, or equivalently if u is a multiple of [1, 1]. It is left as an exercise for the reader to verify that v = [1, 2] is an eigenvector for λ = 1 by showing that this v is a solution to Av v or (A - I)v = 0. (i) [ ] (ii) [4] 23. (a) Compute the eigenvalues of each of the following matrices. 4 (iii) [ ] 園 3 0 2 1 (iv) (v) -1 -3 -2 2 3 (b) Determine an eigenvector associated with the largest eigenvalue, using the method in Example 4, for the matrices in part (a).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.5: Iterative Methods For Computing Eigenvalues
Problem 15EQ
Related questions
Question
100%
not a graded assignment, please do (i) and (iii) for a and b.
![Example 4. Determining Eigenvalues
and Eigenvectors
Consider the system of computer-dog growth equations from Section
2.5.
C'=3C + D or c' = Ac, where A =
D' = 2C + 2D
In Section 2.5 the eigenvalues and eigenvectors were given without
any explanation of how they were found. Let us calculate them now.
By Theorem 2 the eigenvalues are the zeros of the characteristic poly-
nomial det(A-AI):
det(AAI)
=
3-A
2
1
2
-
A
=(3A)(2A)-1-2
= =(65+2)- 2
= 4 5A + A²
= (4-x)(1-x)
So the zeros of det(AAI) = (4A)(1A) are 4 and 1.
(19)
To find an eigenvector u for the eigenvalue 4, we must solve the
system Au=4u or, by matrix algebra, (A-41)u = 0, where
3-4
1
A-41-
=
=
2
2
-
We find that
2n='n 0= n + 'n-
2u₁
-
24₂ = 0
The second equation here is just -2 times the first equation (so it is
superfluous). Then u is an eigenvector if u₁ = u₂, or equivalently if
u is a multiple of [1, 1].
It is left as an exercise for the reader to verify that v = [1, 2]
is an eigenvector for λ = 1 by showing that this v is a solution to
Av v or (A - I)v = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a34a1e4-7c25-4c64-ab7e-d5dd0d192579%2Fe7040310-9737-4506-9c74-1775400ab3a3%2F0lufof_processed.png&w=3840&q=75)
Transcribed Image Text:Example 4. Determining Eigenvalues
and Eigenvectors
Consider the system of computer-dog growth equations from Section
2.5.
C'=3C + D or c' = Ac, where A =
D' = 2C + 2D
In Section 2.5 the eigenvalues and eigenvectors were given without
any explanation of how they were found. Let us calculate them now.
By Theorem 2 the eigenvalues are the zeros of the characteristic poly-
nomial det(A-AI):
det(AAI)
=
3-A
2
1
2
-
A
=(3A)(2A)-1-2
= =(65+2)- 2
= 4 5A + A²
= (4-x)(1-x)
So the zeros of det(AAI) = (4A)(1A) are 4 and 1.
(19)
To find an eigenvector u for the eigenvalue 4, we must solve the
system Au=4u or, by matrix algebra, (A-41)u = 0, where
3-4
1
A-41-
=
=
2
2
-
We find that
2n='n 0= n + 'n-
2u₁
-
24₂ = 0
The second equation here is just -2 times the first equation (so it is
superfluous). Then u is an eigenvector if u₁ = u₂, or equivalently if
u is a multiple of [1, 1].
It is left as an exercise for the reader to verify that v = [1, 2]
is an eigenvector for λ = 1 by showing that this v is a solution to
Av v or (A - I)v = 0.
![(i)
[ ]
(ii)
[4]
23. (a) Compute the eigenvalues of each of the following matrices.
4
(iii)
[ ]
園
3
0 2
1
(iv)
(v)
-1 -3
-2
2
3
(b) Determine an eigenvector associated with the largest eigenvalue,
using the method in Example 4, for the matrices in part (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a34a1e4-7c25-4c64-ab7e-d5dd0d192579%2Fe7040310-9737-4506-9c74-1775400ab3a3%2Fatpgzr_processed.png&w=3840&q=75)
Transcribed Image Text:(i)
[ ]
(ii)
[4]
23. (a) Compute the eigenvalues of each of the following matrices.
4
(iii)
[ ]
園
3
0 2
1
(iv)
(v)
-1 -3
-2
2
3
(b) Determine an eigenvector associated with the largest eigenvalue,
using the method in Example 4, for the matrices in part (a).
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