Rayleigh-Quotient

The Rayleigh quotient, also called Rayleigh coefficient, is an object from linear algebra, which is named after the physicist John William Strutt, 3rd Baron Rayleigh named. The Rayleigh quotient is especially used for the numerical calculation of eigenvalues ​​of a square matrix.

Definition

Is an n-dimensional vector space, a Vektorraumendomorphismus illustrating matrix and a vector, the Rayleigh ratio is defined by

Here denotes the Hermitian transpose of. The image area of the Rayleigh quotient is exactly the numerical range of.

Properties

On multiplication of the vector by a scalar, the Rayleigh quotient does not change: so it is a homogeneous function of degree 0

The Rayleigh quotient has a close relationship to the eigenvalues ​​of. Is an eigenvector of the matrix, and the associated eigenvalue, then:

By the Rayleigh quotient so each eigenvector is imaged by the corresponding eigenvalue. This property is used among others in the numerical calculation of eigenvalues ​​. In particular, for a hermitian matrix with the smallest eigenvalue and the largest eigenvalue:

The calculation of the smallest and largest eigenvalue is thus equivalent to finding the minimum or maximum of the Rayleigh quotient. This can also generalize under appropriate conditions on 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 for asymmetric matrices. Therefore led Ostrowski 1958/59 the so-called two - sided Rayleigh quotient

, where, in turn, is stationary on the right and left eigenvectors. As for normal matrices match right and left eigenvectors of the 2-sided coincides in this case with the ( one-sided ) Rayleigh quotient.

Use in numerical mathematics

For numerical methods for solving eigenvalue problems which, such as the power iteration or inverse iteration, primarily calculate eigenvector approximations, can be additionally using the Rayleigh quotient also eigenvalue approximations determine. In the inverse iteration is a parameter of the shift, are required. Is selected in each iteration as Rayleigh quotient of the current Eigenvektornäherung, this results in the so-called Rayleigh quotient method.