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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 Berezinian ： ウィキペディア英語版 Berezinian In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold. ==Definition== The Berezinian is uniquely determined by two defining properties: *$\backslash operatorname(XY)\; =\; \backslash operatorname(X)\backslash operatorname(Y)$ * Such a matrix is invertible if and only if both ''A'' and ''D'' are invertible matrices over ''K''. The Berezinian of ''X'' is given by :$\backslash operatorname(X)\; =\; \backslash det(A)\backslash det(D)^$ For a motivation of the negative exponent see the substitution formula in the odd case. More generally, consider matrices with entries in a supercommutative algebra ''R''. An even supermatrix is then of the form : where ''A'' and ''D'' have even entries and ''B'' and ''C'' have odd entries. Such a matrix is invertible if and only if both ''A'' and ''D'' are invertible in the commutative ring ''R''0 (the even subalgebra of ''R''). In this case the Berezinian is given by :$\backslash operatorname(X)\; =\; \backslash det(A-BD^C)\backslash det(D)^$ or, equivalently, by :$\backslash operatorname(X)\; =\; \backslash det(A)\backslash det(D-CA^B)^.$ These formulas are well-defined since we are only taking determinants of matrices whose entries are in the commutative ring ''R''0. The matrix : $D-CA^B\; \backslash ,$ is known as the Schur complement of ''A'' relative to An odd matrix ''X'' can only be invertible if the number of even dimensions equals the number of odd dimensions. In this case, invertibility of ''X'' is equivalent to the invertibility of ''JX'', where : Then the Berezinian of ''X'' is defined as :$\backslash operatorname(X)\; =\; \backslash operatorname(JX)\; =\; \backslash det(C-DB^A)\backslash det(-B)^.$ 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「Berezinian」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.