Rayleigh quotient

In mathematics, for a given complex Hermitian matrix M and nonzero vector x, 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 non-zero real scalar c. 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 M) when x is (the corresponding eigenvector). Similarly, and .