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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 Quasideterminant ： ウィキペディア英語版 Quasideterminant In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows: : $$ \left|\begin a_ & a_ \\ a_ & a_ \end \right|_ = a_ - a_a_ \qquad \left|\begin a_ & a_ \\ a_ & a_ \end \right|_ = a_ - a_a_. In general, there are ''n''2 quasideterminants defined for an ''n'' × ''n'' matrix (one for each position in the matrix), but the presence of the inverted terms above should give the reader pause: they are not always defined, and even when they are defined, they do not reduce to determinants when the entries commute. Rather, : $$ \left|A\right|_ = (-1)^ \frac means delete the ''i''th row and ''j''th column from ''A''. The $2\backslash times2$ examples above were introduced between 1926 and 1928 by Richardson 〔A.R. Richardson, Hypercomplex determinants, ''Messenger of Math.'' 55 (1926), no. 1.〕 〔A.R. Richardson, Simultaneous linear equations over a division algebra, ''Proc. London Math. Soc.'' 28 (1928), no. 2.〕 and Heyting, 〔A. Heyting, Die theorie der linearen gleichungen in einer zahlenspezies mit nichtkommutativer multiplikation, ''Math. Ann. 98'' (1928), no. 1.〕 but they were marginalized at the time because they were not polynomials in the entries of $A$. These examples were rediscovered and given new life in 1991 by I.M. Gelfand and V.S. Retakh. 〔I. Gelfand, V. Retakh, Determinants of matrices over noncommutative rings, ''Funct. Anal. Appl.'' 25 (1991), no. 2.〕 〔I. Gelfand, V. Retakh, Theory of noncommutative determinants, and characteristic functions of graphs, ''Funct. Anal. Appl.'' 26 (1992), no. 4.〕 There, they develop quasideterminantal versions of many familiar determinantal properties. For example, if $B$ is built from $A$ by rescaling its $i$-th row (on the left) by $\backslash left.\backslash rho\backslash right.$, then $\backslash left|B\backslash right|\_\; =\; \backslash rho\; \backslash left|A\backslash right|\_$. Similarly, if $B$ is built from $A$ by adding a (left) multiple of the $k$-th row to another row, then $\backslash left|B\backslash right|\_\; =\; \backslash left|A\backslash right|\_\; \backslash ,\backslash ,\; (\backslash forall\; j;\; \backslash forall\; k\backslash neq\; i)$. They even develop a quasideterminantal version of Cramer's rule. ==Definition== Let $A$ be an $n\backslash times\; n$ matrix over a (not necessarily commutative) ring $R$ and fix $1\backslash leq\; i,j\backslash leq\; n$. Let $a\_$ denote the ($i,j$)-entry of $A$, let $r\_i^j$ denote the $i$-th row of $A$ with column $j$ deleted, and let $c\_j^i$ denote the $j$-th column of $A$ with row $i$ deleted. The ($i,j$)-quasideterminant of $A$ is defined if the submatrix $A^$ is invertible over $R$. In this case, ::$$ \left|A\right|_ = a_ - r_i^j\, \bigl(A^\bigr)^\, c_j^i . Recall the formula (for commutative rings) relating $A^$ to the determinant, namely $(A^)\_\; =\; (-1)^\; \backslash frac$. The above definition is a generalization in that (even for noncommutative rings) one has ::$$ \bigl(A^\bigr)_ = \left|A\right|_^ whenever the two sides makes sense. 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「Quasideterminant」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.