Classical groups {symmetric bilinear form} {orthogonal group} can preserve linear vector transformations in three-dimensional Euclidean space, because axes stay at right angles and maintain non-singular quadratic form.
matrices
Matrices can represent orthogonal groups. For orthogonal-group transformations, inverse matrix equals transpose matrix.
Matrix diagonal has 0, 1, or -1. Orthogonal groups {Lorentz group} can have diagonal {Lorentzian diagonal} with one 1, zero 0s, and non-zero -1s. Number of 1s and -1s can be invariant {signature, group}.
If diagonal has only 1s {positive-definite diagonal}, matrix is non-singular and positive, and has positive-definite non-singular symmetric tensors {metric, tensor}. Group {pseudo-orthogonal group} matrix can be singular and not positive-definite {pseudometric}.
If orthogonal-group transformation determinant equals 1, group is non-reflective.
Mathematical Sciences>Group Theory>Mathematical Groups
3-Group Theory-Mathematical Groups
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Date Modified: 2022.0224