Suppose the 3 × 3 matrix A has eigenvalues λ1 = −6, λ2 = −2, and λ3 = 5.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.4: The Singular Value Decomposition
Problem 26EQ
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Suppose the 3 × 3 matrix A has eigenvalues λ1 = −6, λ2 = −2, and λ3 = 5.
![Let v be randomly chosen vector in R³. Set
and for k> 0 set
Ik+1 =
xo
V
||v|| '
(A +41)-¹k
||(A+41)-¹||'
Xo = xo (A +41)-¹xo
Ak
Ak+1 = Xk+1¹(A +41)-¹Xk+1
i. What are the eigenvalues of A + 4I?
ii. What are the eigenvalues of (A +41)−¹?
iii. What number do you expect to converge to?
iv. What number do you expect
4 to converge to?
v. For large k, æk is approximately an eigenvector corresponding to which eigenvalue of A?
37
(6)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fffc06da2-7bd8-42a3-b8a1-3f8b831a8034%2F11f86de3-0283-4934-8985-5fc03618040d%2Fpdxu9ni_processed.png&w=3840&q=75)
Transcribed Image Text:Let v be randomly chosen vector in R³. Set
and for k> 0 set
Ik+1 =
xo
V
||v|| '
(A +41)-¹k
||(A+41)-¹||'
Xo = xo (A +41)-¹xo
Ak
Ak+1 = Xk+1¹(A +41)-¹Xk+1
i. What are the eigenvalues of A + 4I?
ii. What are the eigenvalues of (A +41)−¹?
iii. What number do you expect to converge to?
iv. What number do you expect
4 to converge to?
v. For large k, æk is approximately an eigenvector corresponding to which eigenvalue of A?
37
(6)
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