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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 Diagonal matrix ： ウィキペディア英語版 Diagonal matrix In linear algebra, a diagonal matrix is a matrix (usually a square matrix) in which the entries outside the main diagonal (↘) are all zero. The diagonal entries themselves may or may not be zero. Thus, the matrix with ''n'' columns and ''n'' rows is diagonal if: :$d\_\; =\; 0\; \backslash text\; i\; \backslash ne\; j\backslash \; \backslash forall\; i,j\; \backslash in\; \backslash$ For example, the following matrix is diagonal: :$\backslash begin$ 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -2\end The term ''diagonal matrix'' may sometimes refer to a rectangular diagonal matrix, which is an ''m''-by-''n'' matrix with all the entries not of the form ''d''''i'',''i'' being zero. For example: :$\backslash begin$ 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -3\\ 0 & 0 & 0\\ \end or $\backslash begin$ 1 & 0 & 0 & 0 & 0\\ 0 & 4 & 0& 0 & 0\\ 0 & 0 & -3& 0 & 0\end However, in the remainder of this article we will consider only square matrices. Any square diagonal matrix is also a symmetric matrix. Also, if the entries come from the field R or C, then it is a normal matrix as well. Equivalently, we can define a diagonal matrix as a matrix that is both upper- and lower-triangular. The identity matrix ''I''''n'' and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal. == Scalar matrix == A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple ''λI'' of the identity matrix ''I''. Its effect on a vector is scalar multiplication by ''λ''. For example, a 3×3 scalar matrix has the form: :$$ \begin \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end \equiv \lambda \boldsymbol_3 The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size. For an abstract vector space ''V'' (rather than the concrete vector space $K^n$), or more generally a module ''M'' over a ring ''R'', with the endomorphism algebra End(''M'') (algebra of linear operators on ''M'') replacing the algebra of matrices, the analog of scalar matrices are scalar transformations. Formally, scalar multiplication is a linear map, inducing a map $R\; \backslash to\; \backslash operatorname(M),$ (send a scalar ''λ'' to the corresponding scalar transformation, multiplication by ''λ'') exhibiting End(''M'') as a ''R''-algebra. For vector spaces, or more generally free modules $M\; \backslash cong\; R^n$, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the center of the endomorphism algebra, and similarly invertible transforms are the center of the general linear group GL(''V''), where they are denoted by Z(''V''), follow the usual notation for the center. 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「Diagonal matrix」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.