Rayleigh quotient

Definition.

The Rayleigh quotient, $R_{\mathrm{A}}$, to the Hermitian matrix $\mathrm{A}$ is defined as

$$R_{𝐀}(𝐱)=\frac{{𝐱}^H\mathrm{𝐀𝐱}}{{𝐱}^H𝐱},𝐱\ne \mathrm{𝟎},$$

where ${\mathrm{x}}^H$ is the Hermitian conjugate of $\mathrm{x}$.

The importance of this quantity (in fact, the reason Rayleigh firstintroduced it) is that its critical values are the eigenvectorsof $A$ and the values of the quotient at these special vectors are thecorresponding eigenvalues. This observation leads to the variationalmethod for computing the spectrum of a positive matrix (either exactly orapproximately). Namely, one first minimizes the Rayleigh quotient over thewhole vector space. This gives the lowest eigenvalue and correspondingeigenvector. Next, one restricts attention to the orthogonal complementof the eigenvector found in the first step and minimizes over this subspace.That produces the next lowest eigenvalue and corresponding eigenvector. Onecan continue this process recursively. At each step, one minimizes theRayleigh quotient over the subspace orthogonal to all the vectors found inthe preceding steps to find another eigenvalue and its correspondingeigenvector.

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