Rayleigh quotient

In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient $R(M, x)$ , is defined as:

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 $R(M, c x) = R(M,x)$ 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 $\lambda_\min$ (the smallest eigenvalue of M) when x is $v_\min$ (the corresponding eigenvector). Similarly, $R(M, x) \leq \lambda_\max$ and $R(M, v_\max) = \lambda_\max$ .

单词	Rayleigh quotient
释义	Rayleigh quotient 英语百科 Rayleigh quotient In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient $R(M, x)$ , is defined as: 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 $R(M, c x) = R(M,x)$ 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 $\lambda_\min$ (the smallest eigenvalue of M) when x is $v_\min$ (the corresponding eigenvector). Similarly, $R(M, x) \leq \lambda_\max$ and $R(M, v_\max) = \lambda_\max$ .
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