Rayleigh Quotient

In mathematics, for a given complex Hermitian matrix and nonzero vector, the Rayleigh quotient, is defined as:

For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose to the usual transpose . Note that for any real scalar . Recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value (the smallest eigenvalue of ) when is (the corresponding eigenvector). Similarly, and . The Rayleigh quotient is used in 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 is called a numerical range.