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The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose).
Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA T will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A. An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices.
The indefinite special orthogonal group, SO(p, q) is the subgroup of O(p, q) consisting of all elements with determinant 1. Unlike in the definite case, SO( p , q ) is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO + ( p , q ) and O + ( p , q ) , which has 2 components ...
Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way: (+) where A is an orthogonal matrix. or the same orthogonal transformation followed by a translation: +, with c = Ab. T(n) is a normal subgroup of E(n): for every translation t and every isometry u, the composition ...
In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization).
In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group , defined by the relation M T M = I n {\displaystyle M^{T}M=I_{n}} where M T {\displaystyle M^{T}} is the transpose of M {\displaystyle M} .
A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix R, note that det R T = det R implies (det R) 2 = 1, so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3).