Editing Rayleigh quotient (section)

{{Short description|Construct for Hermitian matrices}}
{{Use American English|date = April 2019}}
In [[mathematics]], the '''Rayleigh quotient'''<ref>Also known as the '''Rayleigh–Ritz ratio'''; named after [[Walther Ritz]] and [[Lord Rayleigh]].</ref> ({{IPAc-en|ˈ|r|eɪ|.|l|i}}) for a given complex [[Hermitian matrix]] <math>M</math> and nonzero [[vector (geometry)|vector]] ''<math>x</math>'' is defined as:<ref>{{cite book |last1=Horn |first1=R. A. |author1-link=Roger Horn |first2=C. A. |last2=Johnson |author2-link=Charles Royal Johnson |year=1985 |title=Matrix Analysis |publisher=Cambridge University Press |pages=176–180 |isbn=0-521-30586-1 |url=https://books.google.com/books?id=PlYQN0ypTwEC&pg=PA176 }}</ref><ref>{{cite book |last=Parlett |first=B. N. |authorlink=Beresford Parlett |title=The Symmetric Eigenvalue Problem |publisher=SIAM |series=Classics in Applied Mathematics |year=1998 |isbn=0-89871-402-8 }}</ref><math display="block">R(M,x) = {x^{*} M x \over x^{*} x}.</math>For real matrices and vectors, the condition of being Hermitian reduces to that of being [[Symmetric matrix|symmetric]], and the [[conjugate transpose]] <math>x^{*}</math> to the usual [[transpose]] <math>x'</math>. Note that <math>R(M, c x) = R(M,x)</math> for any non-zero scalar ''<math>c</math>''. Recall that a Hermitian (or real symmetric) matrix is [[spectral theorem|diagonalizable with only real eigenvalues]]. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value <math>\lambda_\min</math> (the smallest [[eigenvalue]] of ''<math>M</math>'') when ''<math>x</math>'' is <math>v_\min</math> (the corresponding [[eigenvector]]).<ref>{{cite web |first=Rodica D. |last=Costin |date=2013 |title=Midterm notes |work=Mathematics 5102 Linear Mathematics in Infinite Dimensions, lecture notes |publisher=The Ohio State University |url=https://people.math.osu.edu/costin.10/5102/Rayleigh%20quotient.pdf }}</ref> Similarly, <math>R(M, x) \leq \lambda_\max</math> and <math>R(M, v_\max) = \lambda_\max</math>.

The Rayleigh quotient is used in the [[min-max theorem]] to get exact values of all eigenvalues. It is also used in [[eigenvalue algorithm]]s (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_(functional_analysis)|spectrum]]. When the matrix is Hermitian, the numerical radius is equal to the spectral norm. Still in functional analysis, <math>\lambda_\max</math> is known as the [[spectral radius]]. In the context of <math>C^\star</math>-algebras or algebraic quantum mechanics, the function that to ''<math>M</math>'' associates the Rayleigh–Ritz quotient  <math>R(M, x)</math> for a fixed ''<math>x</math>'' and ''<math>M</math>'' varying through the algebra would be referred to as ''vector state'' of the algebra.

In [[quantum mechanics]], the Rayleigh quotient gives the [[expectation value (quantum mechanics) |expectation value]] of the observable corresponding to the operator ''<math>M</math>'' for a system whose state is given by ''<math>x</math>''.

If we fix the complex matrix ''<math>M</math>'', then the resulting Rayleigh quotient map (considered as a function of ''<math>x</math>'') completely determines ''<math>M</math>'' via the [[polarization identity#Complex numbers|polarization identity]]; indeed, this remains true even if we allow ''<math>M</math>'' to be non-Hermitian. However, if we restrict the field of scalars to the real numbers, then the Rayleigh quotient only determines the [[symmetric matrix|symmetric]] part of ''<math>M</math>''.