In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Rayleigh quotient () for a given complex
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
''M'' and nonzero
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
''x'' is defined as:
For real matrices and vectors, the condition of being Hermitian reduces to that of being
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, and the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
to the usual
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
. 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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of ''M'') when ''x'' is
(the corresponding
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
).
Similarly,
and
.
The Rayleigh quotient is used in the
min-max theorem
In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators o ...
to get exact values of all eigenvalues. It is also used in
eigenvalue algorithm
In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.
Eigenvalues and eigenvectors
Given an square ...
s (such as
Rayleigh quotient iteration Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates.
Rayleigh quotient iteration is an iterative method, that is, ...
) 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 In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex n \times n matrix ''A'' is the set
:W(A) = \left\
where \mathbf^* denotes the conjugate transpose of the vector \mathbf. The nume ...
and contains its
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
. 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 mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
. In the context of C*-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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the Rayleigh quotient gives the
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
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
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product then t ...
; 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
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
part of ''M''.)
Bounds for Hermitian ''M''
As stated in the introduction, for any vector ''x'', one has