In mathematics, the Rayleigh quotient[1] (/ˈreɪ.li/) for a given complex Hermitian matrix
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose
It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value
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.
-algebras or algebraic quantum mechanics, the function that to
In quantum mechanics, the Rayleigh quotient gives the expectation value of the observable corresponding to the operator
, then the resulting Rayleigh quotient map (considered as a function of
via the polarization identity; indeed, this remains true even if we allow
However, if we restrict the field of scalars to the real numbers, then the Rayleigh quotient only determines the symmetric part of
This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M:
It is then easy to verify that the bounds are attained at the corresponding eigenvectors
has non-negative eigenvalues, and orthogonal (or orthogonalisable) eigenvectors, which can be demonstrated as follows.
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
The last representation establishes that the Rayleigh quotient is the sum of the squared cosines of the angles formed by the vector
This then becomes a linear program, which always attains its maximum at one of the corners of the domain.
(when the eigenvalues are ordered by decreasing magnitude).
Thus, the Rayleigh quotient is maximized by the eigenvector with the largest eigenvalue.
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
Because of this invariance, it is sufficient to study the special case
The problem is then to find the critical points of the function
In other words, it is to find the critical points of
are the critical points of the Rayleigh quotient and their corresponding eigenvalues
This property is the basis for principal components analysis and canonical correlation.
Sturm–Liouville theory concerns the action of the linear operator
of functions satisfying some specified boundary conditions at a and b.