Chapter 6, Problem 12SE

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

  Av = λv (1)

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