Rayleigh quotient
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In mathematics, for a given complex Hermitian matrix A and nonzero vector x, the Rayleigh quotient R(A,x) is defined as:
- <math>{x^{*} A x \over x^{*} x}<math>
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose <math>x^{*}<math> to the usual transpose <math>x^{T}<math>.
Note that R(A,c·x) = R(A,x) for any real scalar c.
Recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown that the Rayleigh quotient reaches its minimum value λmin (the smallest eigenvalue of A) when x is vmin (the corresponding eigenvector). Similarly, R(A,x) ≤ λmax and R(A,vmax) = λmax.
The Rayleigh quotient is used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.