Rayleigh Quotient

The Rayleigh quotient of a Hermitian matrix and a nonzero vector is

(1)

where denotes the conjugate transpose. It is real and is unchanged when is multiplied by a nonzero scalar.

If are the eigenvalues of , then

			(2)
			(3)

To see this, expand in an orthonormal basis of eigenvectors. The quotient becomes the convex combination

(4)

Its stationary values subject to are precisely the eigenvalues, attained at the corresponding eigenvectors.

For example, take the symmetric matrix with and . Its eigenvalues are 1 and 3. For the unit vector , the quotient is

(5)

The figure above marks the eigenvector directions and the sample direction .

For a positive definite matrix , the generalized quotient is

(6)

Its stationary values are the generalized eigenvalues defined by .

For a self-adjoint operator on a Hilbert space, the corresponding expression is

(7)

for nonzero in the operator's domain. If is bounded below and has a lowest eigenvalue , then , with equality at a lowest eigenfunction (Teschl 2009, pp. 118-119). Restricting the trial functions to a finite-dimensional subspace gives the Rayleigh-Ritz variational technique.