Rayleigh quotient: Mathematics, Hermitian Matrix, Conjugate Transpose, Eigenvalue, Eigenvector and Eeigenspace, Min Max Theorem, Quotient Iteration, Numerical Range, Euclidean Vector

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x * to the usual transpose x'. Note that R(A,cx) = R(A,x) for any 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 lambda_{operatorname{min}} (the smallest eigenvalue of A) when x is v_{operatorname{min}} (the corresponding eigenvector). Similarly, R(A, x) leq lambda_{operatorname{max}} and R(A, v_{operatorname{max}}) = lambda_{operatorname{max}}. 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.

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