Rayleigh quotient

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

For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose ${\displaystyle x^{*}}$ to the usual transpose ${\displaystyle x'}$. Note that ${\displaystyle R(M,cx)=R(M,x)}$ 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 ${\displaystyle \lambda _{\min }}$ (the smallest eigenvalue of M) when x is ${\displaystyle v_{\min }}$ (the corresponding eigenvector). Similarly, ${\displaystyle R(M,x)\leq \lambda _{\max }}$ and ${\displaystyle R(M,v_{\max })=\lambda _{\max }}$.

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