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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 Matrix norm ： ウィキペディア英語版 Matrix norm In mathematics, a matrix norm is a natural extension of the notion of a vector norm to matrices. == Definition == In what follows, $K$ will denote the field of real or complex numbers. Let $K^$ denote the vector space containing all matrices with $m$ rows and $n$ columns with entries in $K$. Throughout, $A^$ * denotes the conjugate transpose of matrix $A$. A matrix norm is a vector norm on $K^$. That is, if $\backslash |A\backslash |$ denotes the norm of the matrix $A$, then, *$\backslash |A\backslash |\backslash ge\; 0$ *$\backslash |A\backslash |=\; 0$ iff $A=0$ *$\backslash |\backslash alpha\; A\backslash |=|\backslash alpha|\; \backslash |A\backslash |$ for all $\backslash alpha$ in $K$ and all matrices $A$ in $K^$ *$\backslash |A+B\backslash |\; \backslash le\; \backslash |A\backslash |+\backslash |B\backslash |$ for all matrices $A$ and $B$ in $K^.$ Additionally, in the case of square matrices (thus, ''m'' = ''n''), some (but not all) matrix norms satisfy the following condition, which is related to the fact that matrices are more than just vectors: *$\backslash |AB\backslash |\; \backslash le\; \backslash |A\backslash |\backslash |B\backslash |$ for all matrices $A$ and $B$ in $K^.$ A matrix norm that satisfies this additional property is called a sub-multiplicative norm (in some books, the terminology ''matrix norm'' is used only for those norms which are sub-multiplicative). The set of all ''n''-by-''n'' matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra. 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「Matrix norm」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.