# Rayleigh quotient

###### Definition.

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

$${R}_{\mathbf{A}}(\mathbf{x})=\frac{{\mathbf{x}}^{H}\mathrm{\mathbf{A}\mathbf{x}}}{{\mathbf{x}}^{H}\mathbf{x}},\mathbf{x}\ne \mathrm{\U0001d7ce},$$ |

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

The importance of this quantity (in fact, the reason Rayleigh first
introduced it) is that its critical values are the eigenvectors^{}
of $A$ and the values of the quotient at these special vectors are the
corresponding eigenvalues^{}. 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 space^{}. This gives the lowest eigenvalue and corresponding
eigenvector. Next, one restricts attention to the orthogonal complement^{}
of the eigenvector found in the first step and minimizes over this subspace^{}.
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 orthogonal^{} 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 operator^{} 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 |