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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 Commuting matrices ： ウィキペディア英語版 Commuting matrices In linear algebra, two matrices $A$ and $B$ are said to commute if $AB=BA$ and equivalently, their commutator $()=\; AB-BA$ is zero. A set of matrices $A\_1,\backslash ldots,A\_k$ is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other. == Characterization in terms of eigenvectors == Commuting matrices over an algebraically closed field are simultaneously triangularizable, in other words they will be both upper triangular on a same basis. This follows from the fact that commuting matrices preserve each other's eigenspaces. If both matrices are diagonalizable, then they can be simultaneously diagonalized. Moreover, if one of the matrices has the property that its minimal polynomial coincides with its characteristic polynomial (i.e., it has the maximal degree), which happens in particular whenever the characteristic polynomial has only simple roots, then the other matrix can be written as a polynomial of the first. As a direct consequence of simultaneous triangulizability, the eigenvalues of two commuting matrices complex ''A'', ''B'' with their algebraic multiplicities (the multisets of roots of their characteristic polynomials) can be matched up as $\backslash alpha\_i\backslash leftrightarrow\backslash beta\_i$ in such a way that the multiset of eigenvalues of any polynomial $P(A,B)$ in the two matrices is the multiset of the values $P(\backslash alpha\_i,\backslash beta\_i)$. Two Hermitian matrices commute if their eigenspaces coincide. In particular, two Hermitian matrices without multiple eigenvalues commute if they share the same set of eigenvectors. This follows by considering the eigenvalue decompositions of both matrices. Let $A$ and $B$ be two Hermitian matrices. $A$ and $B$ have common eigenspaces when they can be written as : $A\; =\; U\; \backslash Lambda\_1\; U^\backslash dagger$ : $B\; =\; U\; \backslash Lambda\_2\; U^\backslash dagger$ It then follows that : $AB\; =\; U\; \backslash Lambda\_1\; U^\backslash dagger\; U\; \backslash Lambda\_2\; U^\backslash dagger\; =\; U\; \backslash Lambda\_1\; \backslash Lambda\_2\; U^\backslash dagger\; =\; U\; \backslash Lambda\_2\; \backslash Lambda\_1\; U^\backslash dagger\; =\; U\; \backslash Lambda\_2\; U^\backslash dagger\; U\; \backslash Lambda\_1\; U^\backslash dagger\; =\; BA.$ 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「Commuting matrices」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.