Editing Hermitian matrix (section)

==Rayleigh quotient==
{{Main|Rayleigh quotient}}

In mathematics, for a given complex Hermitian matrix {{mvar|M}} and nonzero vector {{math|'''x'''}}, the Rayleigh quotient<ref>Also known as the '''Rayleigh–Ritz ratio'''; named after [[Walther Ritz]] and [[Lord Rayleigh]].</ref> <math>R(M, \mathbf{x}),</math> is defined as:<ref name="HornJohnson"/>{{rp|p. 234}}<ref>Parlet B. N. ''The symmetric eigenvalue problem'', SIAM, Classics in Applied Mathematics,1998</ref>
<math display=block>R(M, \mathbf{x}) := \frac{\mathbf{x}^\mathsf{H} M \mathbf{x}}{\mathbf{x}^\mathsf{H} \mathbf{x}}.</math>

For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose <math>\mathbf{x}^\mathsf{H}</math> to the usual transpose <math>\mathbf{x}^\mathsf{T}.</math> <math>R(M, c \mathbf x) = R(M, \mathbf x)</math> for any non-zero real scalar <math>c.</math> Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.

It can be shown<ref name="HornJohnson" /> that, for a given matrix, the Rayleigh quotient reaches its minimum value <math>\lambda_\min</math> (the smallest eigenvalue of M) when <math>\mathbf x</math> is <math>\mathbf v_\min</math> (the corresponding eigenvector). Similarly, <math>R(M, \mathbf x) \leq \lambda_\max</math> and <math>R(M, \mathbf 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 algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.

The range of the Rayleigh quotient (for matrix that is 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, <math>\lambda_\max</math> is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to {{math|''M''}} associates the Rayleigh quotient {{math|''R''(''M'', ''x'')}} for a fixed {{math|'''x'''}} and {{math|''M''}} varying through the algebra would be referred to as "vector state" of the algebra.