Solved: Consider the problem of finding an eigenvalue of

Chapter , Problem 12E

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Problem 12E

Consider the problem of finding an eigenvalue of an n × n matrix A when an approximate eigenvector v is known. Since v is not exactly correct, the equation

will probably not have a solution. However,  can be estimated by a least-squares solution when (1) is viewed properly. Think of v as an n × 1 matrix V, think of  as a vector in R1, and denote the vector Av by the symbol b. Then (1) becomes b = V , which may also be written as V  = b. Find the least-squares solution of this system of n equations in the one unknown , and write this solution using the original symbols. The resulting estimate for  is called a Rayleigh quotient. See Exercises 11 and 12 in Section 5.8.

Reference Exercises 11 in Section 5.8:

[M] Exercises 7–12 require MATLAB or other computational aid. In Exercises 7 and 8, use the power method with the x0 given. List

In Exercises 9 and 10,

Another estimate can be made for an eigenvalue when an approximate eigenvector is available. Observe that if and the Rayleigh quotient

Reference Exercises 12 in Section 5.8: