翻訳と辞書 Words near each other ・ Specchia ・ Specchia Gallone ・ Specchiolla ・ Speccy ・ SpecDrum ・ SPECfp ・ Specfs ・ SpecGram ・ Spechbach ・ Spechbach-le-Bas ・ Spechbach-le-Haut ・ Spechhorn ・ Specht ・ Specht module ・ Specht v. Netscape Communications Corp. ・ Specht's theorem ・ Spechtenköpfe ・ Spechtensee ・ Spechts Ferry Formation ・ Spechtshorn ・ Special ・ Special (album) ・ Special (film) ・ Special (Garbage song) ・ Special (lighting) ・ Special (Lost) ・ Special 1 TV ・ Special 26 ・ Special 301 Report ・ Special abelian subgroup Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Specht's theorem ： ウィキペディア英語版 Specht's theorem In mathematics, Specht's theorem gives a necessary and sufficient condition for two matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940. Two matrices ''A'' and ''B'' are said to be ''unitarily equivalent'' if there exists a unitary matrix ''U'' such that ''B'' = ''U''  *''AU''.〔, Definition 2.2.1〕 Two matrices which are unitarily equivalent are also similar. Two similar matrices represent the same linear map, but with respect to a different basis; unitary equivalence corresponds to a change from an orthonormal basis to another orthonormal basis. If ''A'' and ''B'' are unitarily equivalent, then tr ''AA'' * = tr ''BB'' *, where tr denotes the trace (in other words, the Frobenius norm is a unitary invariant). This follows from the cyclic invariance of the trace: if ''B'' = ''U''  *''AU'', then tr ''BB'' * = tr ''U''  *''AUU''  *''A'' *''U'' = tr ''AUU''  *''A'' *''UU''  * = tr ''AA'' *, where the second equality is cyclic invariance.〔, Theorem 2.2.2〕 Thus, tr ''AA'' * = tr ''BB'' * is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem gives infinitely many necessary conditions which together are also sufficient. The formulation of the theorem uses the following definition. A word in two variables, say ''x'' and ''y'', is an expression of the form :$$ W(x,y) = x^ y^ x^ y^ \cdots x^, \, where ''m''1, ''n''1, ''m''2, ''n''2, …, ''m''''p'' are non-negative integers. The ''degree'' of this word is :$$ m_1 + n_1 + m_2 + n_2 + \cdots + m_p. \, Specht's theorem: Two matrices ''A'' and ''B'' are unitarily equivalent if and only if tr ''W''(''A'', ''A'' *) = tr ''W''(''B'', ''B'' *) for all words ''W''.〔, Theorem 2.2.6〕 The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Let ''n'' denote the size of the matrices ''A'' and ''B''. For the case ''n'' = 2, the following three conditions are sufficient:〔, Theorem 2.2.8〕 :$$ \operatorname \, A = \operatorname \, B, \quad \operatorname \, A^2 = \operatorname \, B^2, \quad\text\quad \operatorname \, AA^ * = \operatorname \, BB^ *. For ''n'' = 3, the following seven conditions are sufficient: :$$ \begin &\operatorname \, A = \operatorname \, B, \quad \operatorname \, A^2 = \operatorname \, B^2, \quad \operatorname \, AA^ * = \operatorname \, BB^ *, \quad \operatorname \, A^3 = \operatorname \, B^3, \\ &\operatorname \, A^2 A^ * = \operatorname \, B^2 B^ *, \quad \operatorname \, A^2 (A^ *)^2 = \operatorname \, B^2 (B^ *)^2, \quad\text\quad \operatorname \, A^2 (A^ *)^2 A A^ * = \operatorname \, B^2 (B^ *)^2 B B^ *. \end  〔, p. 260, quoted by 〕 For general ''n'', it suffices to show that tr ''W''(''A'', ''A'' *) = tr ''W''(''B'', ''B'' *) for all words of degree at most :$$ n \sqrt + \frac14} + \frac2 - 2.  〔, Theorem 4.3〕 It has been conjectured that this can be reduced to an expression linear in ''n''.〔, p. 160〕 == Notes == 〔 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「Specht's 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.