Rayleigh Quotient

The Rayleigh quotient is a scalar valued expression in linear algebra defined, for a Hermitian matrix M and a nonzero vector x, as the quadratic form x star M x divided by the squared norm x star x, and it appears throughout mathematics, physics and engineering. The quotient is invariant under scaling of the vector, so multiplying x by any nonzero constant leaves its value unchanged. For a Hermitian matrix, which has only real eigenvalues, the Rayleigh quotient reaches its minimum at the matrixs smallest eigenvalue and its maximum at the largest, and equals the corresponding eigenvalue exactly when x is that eigenvalues eigenvector, making the quotient a kind of weighted average of all the matrixs eigenvalues. This extremal behavior underlies the min-max theorem in spectral analysis and the Rayleigh quotient iteration algorithm for numerically refining eigenvalue estimates, and in quantum mechanics the same quotient represents the expectation value of an observable for a given quantum state. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/