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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 Vandermonde matrix ： ウィキペディア英語版 Vandermonde matrix In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an ''m'' × ''n'' matrix :$V=\backslash begin$ 1 & \alpha_1 & \alpha_1^2 & \dots & \alpha_1^\\ 1 & \alpha_2 & \alpha_2^2 & \dots & \alpha_2^\\ 1 & \alpha_3 & \alpha_3^2 & \dots & \alpha_3^\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 1 & \alpha_m & \alpha_m^2 & \dots & \alpha_m^ \end or :$V\_\; =\; \backslash alpha\_i^\; \backslash ,$ for all indices ''i'' and ''j''.〔Roger A. Horn and Charles R. Johnson (1991), ''Topics in matrix analysis,'' Cambridge University Press. ''See Section 6.1''〕 (Some authors use the transpose of the above matrix.) The determinant of a square Vandermonde matrix (where ''m'' = ''n'') can be expressed as: :$\backslash det(V)\; =\; \backslash prod\_\; (\backslash alpha\_j-\backslash alpha\_i).$ This is called the Vandermonde determinant or Vandermonde polynomial. If all the numbers $\backslash alpha\_i$ are distinct, then it is non-zero. The Vandermonde determinant is sometimes called the discriminant, although many sources, including this article, refer to the discriminant as the square of this determinant. Note that the Vandermonde determinant is ''alternating'' in the entries, meaning that permuting the $\backslash alpha\_i$ by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the determinant. It thus depends on the order, while its square (the discriminant) does not depend on the order. When two or more α''i'' are equal, the corresponding polynomial interpolation problem (see below) is underdetermined. In that case one may use a generalization called confluent Vandermonde matrices, which makes the matrix non-singular while retaining most properties. If α''i'' = α''i'' + 1 = ... = α''i''+''k'' and α''i'' ≠ α''i'' − 1, then the (''i'' + ''k'')th row is given by : The above formula for confluent Vandermonde matrices can be readily derived by letting two parameters $\backslash alpha\_i$ and $\backslash alpha\_j$ go arbitrarily close to each other. The difference vector between the rows corresponding to $\backslash alpha\_i$ and $\backslash alpha\_j$ scaled to a constant yields the above equation (for ''k'' = 1). Similarly, the cases ''k'' > 1 are obtained by higher order differences. Consequently, the confluent rows are derivatives of the original Vandermonde row. ==Properties== In the case of a square Vandermonde matrix, the Leibniz formula for the determinant gives :$\backslash det(V)\; =\; \backslash sum\_\; \backslash sgn(\backslash sigma)\; \backslash prod\_^n\; \backslash alpha\_i^,$ where ''S''''n'' denotes the set of permutations of $\backslash$, and $\backslash sgn(\backslash sigma)$ denotes the signature of the permutation ''σ''. This determinant factors as :$\backslash sum\_\; \backslash sgn(\backslash sigma)\; \backslash prod\_^n\; \backslash alpha\_i^=\backslash prod\_\; (\backslash alpha\_j-\backslash alpha\_i).$ Each of these factors must divide the determinant, because the latter is an alternating polynomial in the ''n'' variables. It also follows that the Vandermonde determinant divides any other alternating polynomial; the quotient will be a symmetric polynomial. If ''m'' ≤ ''n'', then the matrix ''V'' has maximum rank (''m'') if and only if all α''i'' are distinct. A square Vandermonde matrix is thus invertible if and only if the α''i'' are distinct; an explicit formula for the inverse is known.〔(Inverse of Vandermonde Matrix (ProofWiki) )〕 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「Vandermonde 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.