Rayleigh quotient - Definition and Overview

In mathematics, for a given real symmetric matrix A and real nonzero vector x, the Rayleigh quotient R(A,x) is defined as:

<math>{x^{T} A x \over x^{T} x}<math>

Note that R(A,c·x) = R(A,x) for any real scalar c.

It can be shown that this quotient reaches its minimum value λ_min (the smallest eigenvalue of A) when x is v_min (the corresponding eigenvector). Similarly, R(A,x) ≤ λ_max and R(A,v_max) = λ_max

The Rayleigh quotient is used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.