Rayleigh quotient

In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient R ( M , x ) {displaystyle R(M,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 ∗ {displaystyle x^{*}} to the usual transpose x ′ {displaystyle x'} . Note that R ( M , c x ) = R ( M , x ) {displaystyle R(M,cx)=R(M,x)} 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 λ min {displaystyle lambda _{min }} (the smallest eigenvalue of M) when x is v min {displaystyle v_{min }} (the corresponding eigenvector). Similarly, R ( M , x ) ≤ λ max {displaystyle R(M,x)leq lambda _{max }} and R ( M , v max ) = λ max {displaystyle R(M,v_{max })=lambda _{max }} . 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 (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, λ max {displaystyle lambda _{max }} is known as the spectral radius. 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, 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.) As stated in the introduction, for any vector x, one has R ( M , x ) ∈ [ λ min , λ max ] {displaystyle R(M,x)in left} , where λ min , λ max {displaystyle lambda _{min },lambda _{max }} are respectively the smallest and largest eigenvalues of M {displaystyle M} . This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M: where ( λ i , v i ) {displaystyle (lambda _{i},v_{i})} is the i {displaystyle i} th eigenpair after orthonormalization and y i = v i ∗ x {displaystyle y_{i}=v_{i}^{*}x} is the i {displaystyle i} th coordinate of x in the eigenbasis. It is then easy to verify that the bounds are attained at the corresponding eigenvectors v min , v max {displaystyle v_{min },v_{max }} .