Another estimate can be made for an eigenvalue when an ap- proximate cigenvector is available. Observe that if Ax = λx, then xAx = x(x) = (x²x), and the Rayleigh quotient xAX R(x): = xx equals à. If x is close to an eigenvector for A, then this quo- tient is close to A. When A is a symmetric matrix (AT A), the Rayleigh quotient R(x) = (x[ Axx)/(x[xx) will have roughly twice as many digits of accuracy as the scaling fac- tor in the power method. Verify this increased accuracy in Exercises 11 and 12 by computing and R(x) for k = 1. ...4. 11. A = 5 I X() = [3] 0 =
Another estimate can be made for an eigenvalue when an ap- proximate cigenvector is available. Observe that if Ax = λx, then xAx = x(x) = (x²x), and the Rayleigh quotient xAX R(x): = xx equals à. If x is close to an eigenvector for A, then this quo- tient is close to A. When A is a symmetric matrix (AT A), the Rayleigh quotient R(x) = (x[ Axx)/(x[xx) will have roughly twice as many digits of accuracy as the scaling fac- tor in the power method. Verify this increased accuracy in Exercises 11 and 12 by computing and R(x) for k = 1. ...4. 11. A = 5 I X() = [3] 0 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Title: Understanding Eigenvalue Estimation through the Rayleigh Quotient**
**Section: Estimating Eigenvalues with an Approximate Eigenvector**
In linear algebra, when an approximate eigenvector \( x \) for a matrix \( A \) is available, we can make an estimate for the corresponding eigenvalue \( \lambda \).
Consider the scenario where \( Ax = \lambda x \). By manipulating the equation, we have:
\[ x^T A x = x^T (\lambda x) = \lambda (x^T x) \]
This leads us to the concept of the Rayleigh Quotient, denoted as:
\[
R(x) = \frac{x^T A x}{x^T x}
\]
The significant property of the Rayleigh Quotient is that if \( x \) is close to the true eigenvector for \( \lambda \), then \( R(x) \) will be close to \( \lambda \).
**Section: Special Case for Symmetric Matrices**
When the matrix \( A \) is symmetric (i.e., \( A^T = A \)), the accuracy of the Rayleigh Quotient improves significantly. For symmetric matrices, the Rayleigh Quotient \( R(x_k) = \frac{x_k^T A x_k}{x_k^T x_k} \) will yield approximately twice as many accurate digits compared to the scaling factor \( \mu_k \) used in the power method.
**Exercise 11: Practical Verification**
To verify this increased accuracy in real scenarios, consider the following exercise:
Given matrix:
\[ A = \begin{bmatrix} 5 & 2 \\ 2 & 2 \end{bmatrix} \]
And initial vector:
\[ x_0 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \]
Compute the scaling factor \( \mu_k \) and the Rayleigh Quotient \( R(x_k) \) for iterations \( k = 1, \ldots, 4 \).
By completing this exercise, learners will gain hands-on experience with eigenvalue estimation using the Rayleigh Quotient and appreciate its accuracy, especially when applied to symmetric matrices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32d3b985-3b1f-482b-afc0-65b6cf3a51bf%2Fcef45836-b3c9-4153-a54e-90ce04b50083%2Fvf0gnhs_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Eigenvalue Estimation through the Rayleigh Quotient**
**Section: Estimating Eigenvalues with an Approximate Eigenvector**
In linear algebra, when an approximate eigenvector \( x \) for a matrix \( A \) is available, we can make an estimate for the corresponding eigenvalue \( \lambda \).
Consider the scenario where \( Ax = \lambda x \). By manipulating the equation, we have:
\[ x^T A x = x^T (\lambda x) = \lambda (x^T x) \]
This leads us to the concept of the Rayleigh Quotient, denoted as:
\[
R(x) = \frac{x^T A x}{x^T x}
\]
The significant property of the Rayleigh Quotient is that if \( x \) is close to the true eigenvector for \( \lambda \), then \( R(x) \) will be close to \( \lambda \).
**Section: Special Case for Symmetric Matrices**
When the matrix \( A \) is symmetric (i.e., \( A^T = A \)), the accuracy of the Rayleigh Quotient improves significantly. For symmetric matrices, the Rayleigh Quotient \( R(x_k) = \frac{x_k^T A x_k}{x_k^T x_k} \) will yield approximately twice as many accurate digits compared to the scaling factor \( \mu_k \) used in the power method.
**Exercise 11: Practical Verification**
To verify this increased accuracy in real scenarios, consider the following exercise:
Given matrix:
\[ A = \begin{bmatrix} 5 & 2 \\ 2 & 2 \end{bmatrix} \]
And initial vector:
\[ x_0 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \]
Compute the scaling factor \( \mu_k \) and the Rayleigh Quotient \( R(x_k) \) for iterations \( k = 1, \ldots, 4 \).
By completing this exercise, learners will gain hands-on experience with eigenvalue estimation using the Rayleigh Quotient and appreciate its accuracy, especially when applied to symmetric matrices.
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