Rayleigh quotient


The Rayleigh quotient, RA, to the Hermitian matrixMathworldPlanetmath A is defined as


where xH is the Hermitian conjugate of x.

The importance of this quantity (in fact, the reason Rayleigh first introduced it) is that its critical values are the eigenvectorsMathworldPlanetmathPlanetmathPlanetmath of A and the values of the quotient at these special vectors are the corresponding eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath. This observation leads to the variational method for computing the spectrum of a positive matrix (either exactly or approximately). Namely, one first minimizes the Rayleigh quotient over the whole vector spaceMathworldPlanetmath. This gives the lowest eigenvalue and corresponding eigenvector. Next, one restricts attention to the orthogonal complementMathworldPlanetmathPlanetmath of the eigenvector found in the first step and minimizes over this subspacePlanetmathPlanetmath. That produces the next lowest eigenvalue and corresponding eigenvector. One can continue this process recursively. At each step, one minimizes the Rayleigh quotient over the subspace orthogonalMathworldPlanetmath to all the vectors found in the preceding steps to find another eigenvalue and its corresponding eigenvector.

This concept of Rayleigh quotient also makes sense in the more general setting when A is a Hermitian operatorMathworldPlanetmath on a Hilbert space. Furthermore, it is possible to make use of the Rayleigh-Ritz method in cases where the operator has a discrete spectrum bounded from below, such as the Laplace operator on a compact domain. This method is often employed in practise because, in physical applications, one is oftentimes interested in only the lowest eigenvalue or perhaps the first few lowest eigenvalues and not so concerned with the rest of the spectrum.

Title Rayleigh quotient
Canonical name RayleighQuotient
Date of creation 2013-03-22 13:39:17
Last modified on 2013-03-22 13:39:17
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 9
Author alozano (2414)
Entry type Definition
Classification msc 34L15
Defines variational method