Orthogonal group

In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. This is a subgroup of the general linear group GL(n,F). More generally the orthogonal group of a non-singular quadratic form over F is the group of matrices preserving the form.

Related Topics:
Mathematics - Field - Group - Orthogonal matrices - Matrix multiplication - Subgroup - General linear group - Quadratic form

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Every orthogonal matrix has determinant either 1 or −1. The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n,F) known as the special orthogonal group SO(n,F). If the characteristic of F is 2, then 1 = −1, hence O(n,F) and SO(n,F) coincide; otherwise the index of SO(n,F) in O(n,F) is 2. In characteristic 2 and even dimension, many authors define the SO(n,F) differently as the kernel of the Dickson invariant; then it usually has index 2 in O(n,F).

Related Topics:
Determinant - Normal subgroup - Characteristic - Index

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Both O(n,F) and SO(n,F) are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix.

Related Topics:
Algebraic group - Transpose - Inverse

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