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In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')〔A quadratic space is a vector space ''V'' together with a quadratic form ''Q''; the ''Q'' is dropped from notation when it is clear.〕 on the associated projective space P(''V''). Explicitly, the projective orthogonal group is the quotient group :PO(''V'') = O(''V'')/ZO(''V'') = O(''V'')/ where O(''V'') is the orthogonal group of (''V'') and ZO(''V'')= is the subgroup of all orthogonal scalar transformations of ''V'' – these consist of the identity and reflection through the origin. These scalars are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" is because the scalar transformations are the center of the orthogonal group. The projective special orthogonal group, PSO, is defined analogously, as the induced action of the special orthogonal group on the associated projective space. Explicitly: :PSO(''V'') = SO(''V'')/ZSO(''V'') where SO(''V'') is the special orthogonal group over ''V'' and ZSO(''V'') is the subgroup of orthogonal scalar transformations with unit determinant. Here ZSO is the center of SO, and is trivial in odd dimension, while it equals in even dimension – this odd/even distinction occurs throughout the structure of the orthogonal groups. By analogy with GL/SL and GO/SO, the projective orthogonal group is also sometimes called the projective ''general'' orthogonal group and denoted PGO. Like the orthogonal group, the projective orthogonal group can be defined over any field and with varied quadratic forms, though, as with the ordinary orthogonal group, the main emphasis is on the ''real'' ''positive definite'' projective orthogonal group; other fields are elaborated in generalizations, below. Except when mentioned otherwise, in the sequel PO and PSO will refer to the real positive definite groups. Like the spin groups and pin groups, which are covers rather than quotients of the (special) orthogonal groups, the projective (special) orthogonal groups are of interest for (projective) geometric analogs of Euclidean geometry, as related Lie groups, and in representation theory. More intrinsically, the (real positive definite) projective orthogonal group PO can be defined as the isometries of real projective space, while PSO can be defined as the orientation-preserving isometries of real projective space (when the space is orientable; otherwise PSO = PO). == Structure == 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「projective orthogonal group」の詳細全文を読む スポンサード リンク
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