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In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . The dimension of the group is . The indefinite special orthogonal group, is the subgroup of consisting of all elements with determinant 1. Unlike in the definite case, is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected and , which has 2 components – see for definition and discussion. The signature of the form determines the group up to isomorphism; interchanging ''p'' with ''q'' amounts to replacing the metric by its negative, and so gives the same group. If either ''p'' or ''q'' equals zero, then the group is isomorphic to the ordinary orthogonal group O(''n''). We assume in what follows that both ''p'' and ''q'' are positive. The group is defined for vector spaces over the reals. For complex spaces, all groups are isomorphic to the usual orthogonal group , since the transform changes the signature of a form. In even dimension, the middle group is known as the split orthogonal group, and is of particular interest. In odd dimension, split form is the almost-middle group . == Examples == The basic example is the squeeze mappings, which is the group of (the identity component of) linear transforms preserving the unit hyperbola. Concretely, these are the matrices and can be interpreted as ''hyperbolic rotations,'' just as the group SO(2) can be interpreted as ''circular rotations.'' In physics, the Lorentz group is of central importance, being the setting for electromagnetism and special relativity. 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「Indefinite orthogonal group」の詳細全文を読む スポンサード リンク
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