Rayleigh quotient

The Rayleigh quotient , also known as the Rayleigh coefficient , is an object from linear algebra that is named after the physicist John William Strutt, 3rd Baron Rayleigh . The Rayleigh quotient is used in particular for the numerical calculation of eigenvalues ​​of a square matrix . ${\displaystyle A}$

definition

Is a real symmetric or complex Hermitian matrix and with a vector , the Rayleigh quotient is the vector defined by ${\displaystyle A\in {\mathbb {K} }^{n\times n}}$ ${\displaystyle x\in {\mathbb {K} }^{n}}$ ${\displaystyle x\neq 0}$ ${\displaystyle A}$ ${\displaystyle x}$

${\displaystyle R_{A}(x)={\frac {x^{H}Ax}{x^{H}x}}\,.}$

Here denotes the adjoint vector of . The image area of the Rayleigh quotient is exactly the numerical value range of . ${\displaystyle x^{H}={\overline {x}}^{T}}$ ${\displaystyle x}$ ${\displaystyle A}$

properties

When the vector is multiplied by a scalar , the Rayleigh quotient does not change: it is therefore a homogeneous function of degree 0. ${\displaystyle x}$ ${\displaystyle \alpha \neq 0}$ ${\displaystyle R_{A}(\alpha x)=R_{A}(x)}$

The Rayleigh quotient is closely related to the eigenvalues of . If there is an eigenvector of the matrix and the associated eigenvalue, then: ${\displaystyle A}$ ${\displaystyle v}$ ${\displaystyle A}$ ${\displaystyle \lambda }$

${\displaystyle R_{A}(v)={\frac {v^{*}Av}{v^{*}v}}={\frac {v^{*}\lambda v}{v^{*}v}}=\lambda \,.}$

With the Rayleigh quotient, each eigenvector is mapped to the associated eigenvalue . This property is used, among other things, in the numerical calculation of eigenvalues. In particular, for a symmetric or Hermitian matrix with the smallest eigenvalue and the greatest eigenvalue according to Courant-Fischer's theorem : ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle A}$ ${\displaystyle \lambda _{\rm {min}}}$ ${\displaystyle \lambda _{\rm {max}}}$

${\displaystyle \lambda _{\rm {min}}\leq R_{A}(x)\leq \lambda _{\rm {max}}\,.}$

The calculation of the smallest or largest eigenvalue is therefore equivalent to finding the minimum or maximum of the Rayleigh quotient. Under suitable conditions, this can also be generalized to the infinite-dimensional case and is known as the Rayleigh-Ritz principle .

The eigenvectors of Hermitian form the stationary points of the Rayleigh quotient. This does not apply to asymmetrical matrices. That is why Ostrowski used the so-called 2-sided Rayleigh quotient in 1958/59 ${\displaystyle A}$

${\displaystyle R_{A}(x,y)={\frac {y^{*}Ax}{y^{*}x}}}$

a, where , which in turn is stationary on the right and left eigenvectors and . Since right and left eigenvectors match for normal matrices , the 2-sided and the (one-sided) Rayleigh quotient coincide in this case. ${\displaystyle y^{*}x\neq 0}$ ${\displaystyle x}$ ${\displaystyle y}$

Use in numerical mathematics

In numerical methods for solving eigenvalue problems which, such as vector iteration or inverse iteration , primarily calculate eigenvector approximations, the Rayleigh quotient can also be used to determine eigenvalue approximations. With the inverse iteration, a parameter , the shift , is required. If the current eigenvector approximation is selected as the Rayleigh quotient in each iteration step , this results in the so-called Rayleigh quotient method . ${\displaystyle \theta }$ ${\displaystyle \theta }$

Individual evidence

1. ↑ Gisela Engeln-Müllges , Klaus Niederdrenk, Reinhard Wodicka: Numerical algorithms: procedures, examples, applications . Gabler Wissenschaftsverlage, 2010, ISBN 978-3-642-13472-2 , p. 271 ( limited preview in Google Book search).
2. ^ Lloyd N. Trefethen , David Bau: Numerical Linear Algebra. SIAM, 1997, ISBN 978-0-89871-361-9 , pp. 207-210 ( limited preview in Google book search).