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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Rayleigh quotient ： ウィキペディア英語版 Rayleigh quotient In mathematics, for a given complex Hermitian matrix ''M'' and nonzero vector ''x'', the Rayleigh quotient〔Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh.〕 $R(M,\; x)$, is defined as:〔Horn, R. A. and C. A. Johnson. 1985. ''Matrix Analysis''. Cambridge University Press. pp. 176–180.〕〔Parlet B. N. ''The symmetric eigenvalue problem'', SIAM, Classics in Applied Mathematics,1998〕 :$R(M,x)\; :=\; x\}.$ 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 $x\text{'}$. 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 $\backslash lambda\_\backslash min$ (the smallest eigenvalue of ''M'') when ''x'' is $v\_\backslash min$ (the corresponding eigenvector). Similarly, $R(M,\; x)\; \backslash leq\; \backslash lambda\_\backslash max$ and $R(M,\; v\_\backslash max)\; =\; \backslash lambda\_\backslash max$. The Rayleigh quotient is used in the 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. The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range, (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, $\backslash lambda\_\backslash max$ is known as the spectral radius. In the context of C *-algebras or algebraic quantum mechanics, the function that to ''M'' associates the Rayleigh-Ritz quotient ''R(M,x)'' for a fixed ''x'' and ''M'' varying through the algebra would be referred to as "vector state" of the algebra. ==Bounds for Hermitian $M$== As stated in the introduction, it is $R(M,x)\; \backslash in\; \backslash left(\backslash lambda\_\backslash max\; \backslash right)$. This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of ''M'': :$R(M,x)\; =\; x\}\; =\; \backslash frac$ where $(\backslash lambda\_i,\; v\_i)$ is the $i$th eigenpair after orthonormalization and $y\_i\; =\; v\_i^$ * x is the $i$th coordinate of ''x'' in the eigenbasis. It is then easy to verify that the bounds are attained at the corresponding eigenvectors $v\_\backslash min,\; v\_\backslash max$. The fact that the quotient is a weighted average of the eigenvalues can be used to identify the second, the third, ... largest eigenvalues. Let $\backslash lambda\_\; =\; \backslash lambda\_1\; \backslash ge\; \backslash lambda\_2\; \backslash ge\; ...\; \backslash ge\; \backslash lambda\_n$ = \lambda_ be the eigenvalues in decreasing order. If $x$ is constrained to be orthogonal to $v\_1$, in which case $y\_1\; =\; v\_1^$ *x = 0 , then $R(M,x)$ has the maximum $\backslash lambda\_2$, which is achieved when $x\; =\; v\_2$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rayleigh quotient」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.