Previous question: 2b) What are the eigenvalues of (A−σI)−1?Hint: Your answer should involve σ and λj (for j=1,⋯,n). First, find eigenvalues of A−σI. Then take advantage of the fact that eigenvalues of A−1 are 1λj (for j=1,⋯,n).Answer provided 2c)Set o = √₂ + min(|A2 − A3|, |A1 – A2|)/3. If you run power iteration on (A — σ1)−¹, what will the algorithm converge to (assuming that itdoes converge successfully)?Hint: the largest eigenvalue of A is ₁, the largest eigenvalue of A-¹ is the largest eigenvalue of (A − aI) is λ; – a with the farthest X;farthest a, the largest eigenvalue of (A — α¹)−¹ is with the closet X; to a. Think about what the largest eigenvalue of the matrix is inthis question.As a reminder, we assumed |A1| > |A2| > |A3| > · · · > |An|· Step-11NOWep 3(A-61)V₁ = AVI-AV= A₁V₁-66V₁= (-6) V₁.A-6 is Eigen value for CA-64)Similarly 12-6, 3-6,stepilNOWeigen values for (A-61)"A₁-6дот 6 areEigen values of (A-61)¹ㅗ25612-6are

Answered: Image /qna-images/answer/0ea3800a-f155-4326-8d33-0d1873eda10e.jpg

2c) Set σ = √₂ + min(|A2 − A3|, |A1 − A2|)/3. If you run power iteration on (A-oI)-¹, what will the algorithm converge to (assuming that it does converge successfully)? Hint: the largest eigenvalue of A is λ₁, the largest eigenvalue of A-¹ is the largest eigenvalue of (A - αI) is λ; – a with the farthest X; farthest a, the largest eigenvalue of (A — aI)-¹ is with the closet X; to a. Think about what the largest eigenvalue of the matrix is in this question. As a reminder, we assumed |A1| > |A2| > |A3| > · · · > |An|-

2c) Set σ = √₂ + min(|A2 − A3|, |A1 − A2|)/3. If you run power iteration on (A-oI)-¹, what will the algorithm converge to (assuming that it does converge successfully)? Hint: the largest eigenvalue of A is λ₁, the largest eigenvalue of A-¹ is the largest eigenvalue of (A - αI) is λ; – a with the farthest X; farthest a, the largest eigenvalue of (A — aI)-¹ is with the closet X; to a. Think about what the largest eigenvalue of the matrix is in this question. As a reminder, we assumed |A1| > |A2| > |A3| > · · · > |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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Question

Previous question: 2b)

What are the eigenvalues of (A−σI)−1?

Hint: Your answer should involve σ and λj (for j=1,⋯,n). First, find eigenvalues of A−σI. Then take advantage of the fact that eigenvalues of A−1 are 1λj (for j=1,⋯,n).

Answer provided

2c)
Set o = √₂ + min(|A2 − A3|, |A1 – A2|)/3. If you run power iteration on (A — σ1)−¹, what will the algorithm converge to (assuming that it
does converge successfully)?
Hint: the largest eigenvalue of A is ₁, the largest eigenvalue of A-¹ is the largest eigenvalue of (A − aI) is λ; – a with the farthest X;
farthest a, the largest eigenvalue of (A — α¹)−¹ is with the closet X; to a. Think about what the largest eigenvalue of the matrix is in
this question.
As a reminder, we assumed |A1| > |A2| > |A3| > · · · > |An|·

Step-11
NOW
ep 3
(A-61)V₁ = AVI-AV
= A₁V₁-66V₁
= (-6) V₁.
A-6 is Eigen value for CA-64)
Similarly 12-6, 3-6,
stepil
NOW
eigen values for (A-61)
"
A₁-6
дот 6 are
Eigen values of (A-61)¹
ㅗ
256
12-6
are

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2b) What are the eigenvalues of (A - σI)-¹? Hint: Your answer should eigenvalues of A−¹ are involve o and A; (for j = 1, ···, n). First, find eigenvalues of A -oI. Then take advantage of the fact that (for j = 1,...,n).