Consider the problem of finding an eigenvalue of an n X 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, λ can be estimated by a least-squares solution when (1) is viewed properly. Think of v as an n X 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 V λ = 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.