In mathematics, the Rayleigh quotient[1] (/ˈr.li/) for a given complex Hermitian matrix and nonzero vector is defined as:[2][3]For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose to the usual transpose . Note that for any non-zero scalar . Recall that a Hermitian (or real symmetric) matrix is diagonalizable with only real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value (the smallest eigenvalue of ) when is (the corresponding eigenvector).[4] Similarly, and .

The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms (such as Rayleigh quotient iteration) to obtain an eigenvalue approximation from an eigenvector approximation.

The range of the Rayleigh quotient (for any matrix, not necessarily Hermitian) is called a numerical range and contains its spectrum. When the matrix is Hermitian, the numerical radius is equal to the spectral norm. Still in functional analysis, is known as the spectral radius. In the context of -algebras or algebraic quantum mechanics, the function that to associates the Rayleigh–Ritz quotient for a fixed and varying through the algebra would be referred to as vector state of the algebra.

In quantum mechanics, the Rayleigh quotient gives the expectation value of the observable corresponding to the operator for a system whose state is given by .

If we fix the complex matrix , then the resulting Rayleigh quotient map (considered as a function of ) completely determines via the polarization identity; indeed, this remains true even if we allow to be non-Hermitian. However, if we restrict the field of scalars to the real numbers, then the Rayleigh quotient only determines the symmetric part of .

Bounds for Hermitian M

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As stated in the introduction, for any vector x, one has  , where   are respectively the smallest and largest eigenvalues of  . This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M:   where   is the  -th eigenpair after orthonormalization and   is the  th coordinate of x in the eigenbasis. It is then easy to verify that the bounds are attained at the corresponding eigenvectors  .

The fact that the quotient is a weighted average of the eigenvalues can be used to identify the second, the third, ... largest eigenvalues. Let   be the eigenvalues in decreasing order. If   and   is constrained to be orthogonal to  , in which case  , then   has maximum value  , which is achieved when  .

Special case of covariance matrices

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An empirical covariance matrix   can be represented as the product   of the data matrix   pre-multiplied by its transpose  . Being a positive semi-definite matrix,   has non-negative eigenvalues, and orthogonal (or orthogonalisable) eigenvectors, which can be demonstrated as follows.

Firstly, that the eigenvalues   are non-negative:  

Secondly, that the eigenvectors   are orthogonal to one another:   if the eigenvalues are different – in the case of multiplicity, the basis can be orthogonalized.

To now establish that the Rayleigh quotient is maximized by the eigenvector with the largest eigenvalue, consider decomposing an arbitrary vector   on the basis of the eigenvectors  :   where   is the coordinate of   orthogonally projected onto  . Therefore, we have:   which, by orthonormality of the eigenvectors, becomes:  

The last representation establishes that the Rayleigh quotient is the sum of the squared cosines of the angles formed by the vector   and each eigenvector  , weighted by corresponding eigenvalues.

If a vector   maximizes  , then any non-zero scalar multiple   also maximizes  , so the problem can be reduced to the Lagrange problem of maximizing   under the constraint that  .

Define:  . This then becomes a linear program, which always attains its maximum at one of the corners of the domain. A maximum point will have   and   for all   (when the eigenvalues are ordered by decreasing magnitude).

Thus, the Rayleigh quotient is maximized by the eigenvector with the largest eigenvalue.

Formulation using Lagrange multipliers

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Alternatively, this result can be arrived at by the method of Lagrange multipliers. The first part is to show that the quotient is constant under scaling  , where   is a scalar  

Because of this invariance, it is sufficient to study the special case  . The problem is then to find the critical points of the function   subject to the constraint   In other words, it is to find the critical points of   where   is a Lagrange multiplier. The stationary points of   occur at   and  

Therefore, the eigenvectors   of   are the critical points of the Rayleigh quotient and their corresponding eigenvalues   are the stationary values of  . This property is the basis for principal components analysis and canonical correlation.

Use in Sturm–Liouville theory

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Sturm–Liouville theory concerns the action of the linear operator   on the inner product space defined by   of functions satisfying some specified boundary conditions at a and b. In this case the Rayleigh quotient is  

This is sometimes presented in an equivalent form, obtained by separating the integral in the numerator and using integration by parts:  

Generalizations

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  1. For a given pair (A, B) of matrices, and a given non-zero vector x, the generalized Rayleigh quotient is defined as:   The generalized Rayleigh quotient can be reduced to the Rayleigh Quotient   through the transformation   where   is the Cholesky decomposition of the Hermitian positive-definite matrix B.
  2. For a given pair (x, y) of non-zero vectors, and a given Hermitian matrix H, the generalized Rayleigh quotient can be defined as:   which coincides with R(H,x) when x = y. In quantum mechanics, this quantity is called a "matrix element" or sometimes a "transition amplitude".

See also

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References

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  1. ^ Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh.
  2. ^ Horn, R. A.; Johnson, C. A. (1985). Matrix Analysis. Cambridge University Press. pp. 176–180. ISBN 0-521-30586-1.
  3. ^ Parlett, B. N. (1998). The Symmetric Eigenvalue Problem. Classics in Applied Mathematics. SIAM. ISBN 0-89871-402-8.
  4. ^ Costin, Rodica D. (2013). "Midterm notes" (PDF). Mathematics 5102 Linear Mathematics in Infinite Dimensions, lecture notes. The Ohio State University.

Further reading

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