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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 Cauchy matrix ： ウィキペディア英語版 Cauchy matrix In mathematics, a Cauchy matrix, named after Augustin Louis Cauchy, is an ''m''×''n'' matrix with elements ''a''''ij'' in the form :$$ a_=};\quad x_i-y_j\neq 0,\quad 1 \le i \le m,\quad 1 \le j \le n where $x\_i$ and $y\_j$ are elements of a field $\backslash mathcal$, and $(x\_i)$ and $(y\_j)$ are injective sequences (they do not contain repeated elements; elements are ''distinct''). The Hilbert matrix is a special case of the Cauchy matrix, where :$x\_i-y\_j\; =\; i+j-1.\; \backslash ;$ Every submatrix of a Cauchy matrix is itself a Cauchy matrix. == Cauchy determinants == The determinant of a Cauchy matrix is clearly a rational fraction in the parameters $(x\_i)$ and $(y\_j)$. If the sequences were not injective, the determinant would vanish, and tends to infinity if some $x\_i$ tends to $y\_j$. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles: The determinant of a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as :$\backslash det\; \backslash mathbf=$     (Schechter 1959, eqn 4). It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A−1 = B = () is given by :$b\_\; =\; (x\_j\; -\; y\_i)\; A\_j(y\_i)\; B\_i(x\_j)\; \backslash ,$     (Schechter 1959, Theorem 1) where ''A''i(x) and ''B''i(x) are the Lagrange polynomials for $(x\_i)$ and $(y\_j)$, respectively. That is, :$A\_i(x)\; =\; \backslash frac\; \backslash quad\backslash text\backslash quad\; B\_i(x)\; =\; \backslash frac,$ with :$A(x)\; =\; \backslash prod\_^n\; (x-x\_i)\; \backslash quad\backslash text\backslash quad\; B(x)\; =\; \backslash prod\_^n\; (x-y\_i).$ 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「Cauchy 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.