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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 Skew-symmetric matrix ： ウィキペディア英語版 Skew-symmetric matrix In mathematics, and in particular linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix ''A'' whose transpose is also its negative; that is, it satisfies the condition If the entry in the and is ''aij'', i.e. then the skew symmetric condition is For example, the following matrix is skew-symmetric: :$\backslash begin$ 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end. == Properties == We assume that the underlying field is not of characteristic 2: that is, that where 1 denotes the multiplicative identity and 0 the additive identity of the given field. Otherwise, a skew-symmetric matrix is just the same thing as a symmetric matrix. Sums and scalar multiples of skew-symmetric matrices are again skew-symmetric. Hence, the skew-symmetric matrices form a vector space. Its dimension is ''n''(''n''−1)/2. Let Mat''n'' denote the space of matrices. A skew-symmetric matrix is determined by ''n''(''n'' − 1)/2 scalars (the number of entries above the main diagonal); a symmetric matrix is determined by ''n''(''n'' + 1)/2 scalars (the number of entries on or above the main diagonal). Let Skew''n'' denote the space of skew-symmetric matrices and Sym''n'' denote the space of symmetric matrices. If then :$A\; =\; \backslash frac(A\; -\; A^(A\; +\; A^,$ :$\backslash mbox\_n\; =\; \backslash mbox\_n\; \backslash oplus\; \backslash mbox\_n,$ where ⊕ denotes the direct sum. Denote with $\backslash langle\; \backslash cdot,\backslash cdot\; \backslash rangle$ the standard inner product on R''n''. The real ''n''-by-''n'' matrix ''A'' is skew-symmetric if and only if :$\backslash langle\; Ax,y\; \backslash rangle\; =\; -\; \backslash langle\; x,\; Ay\backslash rangle\; \backslash quad\; \backslash forall\; x,y\backslash in\backslash Bbb^n.$ This is also equivalent to $\backslash langle\; x,Ax\; \backslash rangle\; =\; 0$ for all ''x'' (one implication being obvious, the other a plain consequence of $\backslash langle\; x+y,\; A(x+y)\backslash rangle\; =0$ for all x and y). Since this definition is independent of the choice of basis, skew-symmetry is a property that depends only on the linear operator A and a choice of inner product. All main diagonal entries of a skew-symmetric matrix must be zero, so the trace is zero. If is skew-symmetric, ; hence 3x3 skew symmetric matrices can be used to represent cross products as matrix multiplications. 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「Skew-symmetric 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.