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In Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space R''n''. Such involutions are easy to characterize and they can be described geometrically. ==Linear involutions== To give a linear involution is the same as giving an involutory matrix, a square matrix ''A'' such that : where ''I'' is the identity matrix. It is a quick check that a square matrix ''D'' whose elements are all zero off the main diagonal and ±1 on the diagonal, that is, a signature matrix of the form : satisfies (1), i.e. is the matrix of a linear involution. It turns out that all the matrices satisfying (1) are of the form :''A''=''U'' −1''DU'', where ''U'' is invertible and ''D'' is as above. That is to say, the matrix of any linear involution is of the form ''D'' up to a matrix similarity. Geometrically this means that any linear involution can be obtained by taking oblique reflections against any number from 0 through ''n'' hyperplanes going through the origin. (The term ''oblique reflection'' as used here includes ordinary reflections.) One can easily verify that ''A'' represents a linear involution if and only if ''A'' has the form :''A = ±(2P - I)'' for a linear projection ''P''. 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「Affine involution」の詳細全文を読む スポンサード リンク
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