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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 Pseudoinverse ： ウィキペディア英語版 Generalized inverse In mathematics, a generalized inverse of a matrix ''A'' is a matrix that has some properties of the inverse matrix of ''A'' but not necessarily all of them. Formally, given a matrix $A\; \backslash in\; \backslash mathbb^$ and a matrix $A^\; \backslash in\; \backslash mathbb^$, $A^$ is a generalized inverse of $A$ if it satisfies the condition $AA^A\; =\; A$. The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible ones. A generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then this inverse is its unique generalized inverse. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. == Types of generalized inverses == The Penrose conditions are used to define different generalized inverses: for $A\; \backslash in\; \backslash mathbb^$ and $A^\; \backslash in\; \backslash mathbb^,$ AA^= A^ |- | 3.) || $(AA^)^\; =\; AA^$ |- | 4.) || $(A^A)^\; =\; A^A$ . |} If $A^$ satisfies condition (1.), it is a generalized inverse of $A$, if it satisfies conditions (1.) and (2.) then it is a generalized reflexive inverse of $A$, and if it satisfies all 4 conditions, then it is a Moore–Penrose pseudoinverse of $A$. Other various kinds of generalized inverses include * One-sided inverse (left inverse or right inverse) If the matrix ''A'' has dimensions $n\; \backslash times\; m$ and is full rank then use the left inverse if $n\; >\; m$ and the right inverse if * * Left inverse is given by $A\_\; =\; \backslash left(A^\; A\backslash right)^\; A^$, i.e. $A\_\; A\; =\; I\_m$ where $I\_m$ is the $m\; \backslash times\; m$ identity matrix. * * Right inverse is given by $A\_\; =\; A^\; \backslash left(A\; A^\backslash right)^$, i.e. $A\; A\_\; =\; I\_n$ where $I\_n$ is the $n\; \backslash times\; n$ identity matrix. * Drazin inverse * Bott–Duffin inverse * Moore–Penrose pseudoinverse 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「Generalized inverse」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.