In mathematics, the Rayleigh quotient for a given complex Hermitian matrix M and nonzero vector x is defined as: For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x^{*} to the usual transpose x'. Note that for any non-zero scalar c. 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 M) when x is v_\min (the corresponding eigenvector). 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 C^\star-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh–Ritz quotient R(M, x) for a fixed x and M 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 M for a system whose state is given by x. If we fix the complex matrix M, then the resulting Rayleigh quotient map (considered as a function of x) completely determines M via the polarization identity; indeed, this remains true even if we allow M 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 M.

Bounds for Hermitian M

As stated in the introduction, for any vector x, one has, where are respectively the smallest and largest eigenvalues of M. This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M: where is the i-th eigenpair after orthonormalization and is the ith 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 n=2 and x is constrained to be orthogonal to v_1, in which case, then R(M,x) has maximum value \lambda_2, which is achieved when x = v_2.

Special case of covariance matrices

An empirical covariance matrix M can be represented as the product A'A of the data matrix A pre-multiplied by its transpose A'. Being a positive semi-definite matrix, M has non-negative eigenvalues, and orthogonal (or orthogonalisable) eigenvectors, which can be demonstrated as follows. Firstly, that the eigenvalues \lambda_i are non-negative: Secondly, that the eigenvectors v_i 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 x on the basis of the eigenvectors v_i: where is the coordinate of x orthogonally projected onto v_i. 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 x and each eigenvector v_i, weighted by corresponding eigenvalues. If a vector x maximizes R(M,x), then any non-zero scalar multiple kx also maximizes R, 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 i > 1 (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

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 x \to cx, where c 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 \lambda is a Lagrange multiplier. The stationary points of occur at and Therefore, the eigenvectors of M are the critical points of the Rayleigh quotient and their corresponding eigenvalues are the stationary values of \mathcal{L}. This property is the basis for principal components analysis and canonical correlation.

Use in Sturm–Liouville theory

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