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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 Gershgorin circle theorem ： ウィキペディア英語版 Gershgorin circle theorem In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. The spelling of S. A. Gershgorin's name has been transliterated in several different ways, including Geršgorin, Gerschgorin, Gershgorin and Hershhorn/Hirschhorn. ==Statement and proof== Let $A$ be a complex $n\backslash times\; n$ matrix, with entries $a\_\backslash ,$. For $i\; \backslash in\backslash$ let $R\_i\; =\; \backslash sum\_\backslash right|$ be the sum of the absolute values of the non-diagonal entries in the $i$-th row. Let $D(a\_,\; R\_i\; )$ be the closed disc centered at $a\_$ with radius $R\_i$. Such a disc is called a Gershgorin disc. Theorem: Every eigenvalue of $A$ lies within at least one of the Gershgorin discs $D(a\_,R\_i)$ ''Proof'': Let $\backslash lambda$ be an eigenvalue of $A$ and let x = (''x''''j'') be a corresponding eigenvector. Let ''i'' ∈  be chosen so that |''x''''i''| = max''j'' |''x''''j''|. (That is to say, choose i so that xi is the largest (in absolute value) number in the vector x) Then |''x''''i''| > 0, otherwise x = 0. Since x is an eigenvector, $A\; x=\backslash lambda\; x$, and thus: : $\backslash sum\_j\; a\_\; x\_j\; =\; \backslash lambda\; x\_i\; \backslash quad\; \backslash forall\; i\; \backslash in\; \backslash .$ So, splitting the sum, we get : $\backslash sum\_\; a\_\; x\_j\; =\; \backslash lambda\; x\_i\; -\; a\_\; x\_i.$ We may then divide both sides by ''x''''i'' (choosing ''i'' as we explained, we can be sure that ''x''''i'' ≠ 0) and take the absolute value to obtain : $|\backslash lambda\; -\; a\_|\; =\; \backslash left|\backslash frac\; x\_j\}\backslash right|\; \backslash le\; \backslash sum\_\; \backslash left|\; \backslash frac\; \backslash right|\; \backslash le\; \backslash sum\_\; |a\_|\; =\; R\_i$ where the last inequality is valid because : $\backslash left|\; \backslash frac\; \backslash right|\; \backslash leq\; 1\; \backslash quad\; \backslash textj\; \backslash neq\; i.$ Corollary: The eigenvalues of ''A'' must also lie within the Gershgorin discs ''C''''j'' corresponding to the columns of ''A''. ''Proof'': Apply the Theorem to ''A''T. Example For a diagonal matrix, the Gershgorin discs coincide with the spectrum. Conversely, if the Gershgorin discs coincide with the spectrum, the matrix is diagonal. 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「Gershgorin circle theorem」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.