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 =

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.1: Eigenvalues And Eigenvectors

Problem 8E

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