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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 Smith normal form ： ウィキペディア英語版 Smith normal form In mathematics, the Smith normal form is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working with finitely generated modules over a PID, and in particular for deducing the structure of a quotient of a free module. ==Definition== Let ''A'' be a nonzero ''m''×''n'' matrix over a principal ideal domain ''R''. There exist invertible $m\; \backslash times\; m$ and $n\; \backslash times\; n$-matrices ''S, T'' so that the product ''S A T'' is $$ \begin \alpha_1 & 0 & 0 & & \cdots & & 0 \\ 0 & \alpha_2 & 0 & & \cdots & & 0 \\ 0 & 0 & \ddots & & & & 0\\ \vdots & & & \alpha_r & & & \vdots \\ & & & & 0 & & \\ & & & & & \ddots & \\ 0 & & & \cdots & & & 0 \end. and the diagonal elements $\backslash alpha\_i$ satisfy . This is the Smith normal form of the matrix ''A''. The elements $\backslash alpha\_i$ are unique up to multiplication by a unit and are called the ''elementary divisors'', ''invariants'', or ''invariant factors''. They can be computed (up to multiplication by a unit) as : $\backslash alpha\_i\; =\; \backslash frac,$ where $d\_i(A)$ (called ''i''-th ''determinant divisor'') equals the greatest common divisor of all $i\backslash times\; i$ minors of the matrix ''A''. 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「Smith normal form」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.