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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 Angles between flats ： ウィキペディア英語版 Angles between flats The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a plane in space can be generalised to arbitrary dimension. This generalisation was first discussed by Jordan.〔 For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant.〔 These angles are called canonical〔 or principal.〔 The concept of angles can be generalised to pairs of flats in a finite-dimensional inner product space over the complex numbers. ==Jordan's definition〔== Let $F$ and $G$ be flats of dimensions $k$ and $l$ in the $n$-dimensional Euclidean space $E^n$. By definition, a translation of $F$ or $G$ does not alter their mutual angles. If $F$ and $G$ do not intersect, they will do so upon any translation of $G$ which maps some point in $G$ to some point in $F$. It can therefore be assumed without loss of generality that $F$ and $G$ intersect. Jordan shows that Cartesian coordinates $x\_1,\backslash dots,x\_\backslash rho,$ $y\_1,\backslash dots,y\_\backslash sigma,$ $z\_1,\backslash dots,z\_\backslash tau,$ $u\_1,\backslash dots,u\_\backslash upsilon,$ $v\_1,\backslash dots,x\_\backslash alpha,$ $w\_1,\backslash dots,w\_\backslash alpha$ in $E^n$ can then be defined such that $F$ and $G$ are described, respectively, by the sets of equations : $x\_1=0,\backslash dots,x\_\backslash rho=0,$ : $u\_1=0,\backslash dots,u\_\backslash upsilon=0,$ : $v\_1=0,\backslash dots,v\_\backslash alpha=0$ and : $x\_1=0,\backslash dots,x\_\backslash rho=0,$ : $z\_1=0,\backslash dots,z\_\backslash tau=0,$ : $v\_1\backslash cos\backslash theta\_1+w\_1\backslash sin\backslash theta\_1=0,\backslash dots,v\_\backslash alpha\backslash cos\backslash theta\_\backslash alpha+w\_\backslash alpha\backslash sin\backslash theta\_\backslash alpha=0$ with . Jordan calls these coordinates canonical. By definition, the angles $\backslash theta\_i$ are the angles between $F$ and $G$. The non-negative integers $\backslash rho,\backslash sigma,\backslash tau,\backslash upsilon,\backslash alpha$ are constrained by : $\backslash rho+\backslash sigma+\backslash tau+\backslash upsilon+2\backslash alpha=n,$ : $\backslash sigma+\backslash tau+\backslash alpha=k,$ : $\backslash sigma+\backslash upsilon+\backslash alpha=l.$ For these equations to determine the five non-negative integers completely, besides the dimensions $n,k$ and $l$ and the number $\backslash alpha$ of angles $\backslash theta\_i$, the non-negative integer $\backslash sigma$ must be given. This is the number of coordinates $y\_i$, whose corresponding axes are those lying entirely within both $F$ and $G$. The integer $\backslash sigma$ is thus the dimension of $F\backslash cap\; G$. The set of angles $\backslash theta\_i$ may be supplemented with $\backslash sigma$ angles $0$ to indicate that $F\backslash cap\; G$ has that dimension. Jordan's proof applies essentially unaltered when $E^n$ is replaced with the $n$-dimensional inner product space $\backslash mathbb\; C^n$ over the complex numbers. (For angles between subspaces, the generalisation to $\backslash mathbb\; C^n$ is discussed by Galántai and Hegedũs in terms of the below variational characterisation.〔) 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「Angles between flats」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.