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The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3). The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space.
The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1.
The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO(n), one example of which is the rotation group SO(3). The set of all orthogonal matrices of size n with determinant +1 or −1 is a representation of the (general) orthogonal group O(n).
Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system). The basic problem is to specify the orientation of these three unit vectors , and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.
Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigenvalue, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation. If R has more than one invariant vector then φ = 0 and R = I. Any vector is an invariant vector of I.
Examples include the special orthogonal group (which if n is 2 or 3 consists of all rotation matrices), and the special unitary group. Because the determinant respects multiplication and inverses, it is in fact a group homomorphism from GL n ( K ) {\displaystyle \operatorname {GL} _{n}(K)} into the multiplicative group K × {\displaystyle K ...
The set of n × n orthogonal matrices, under multiplication, forms the group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix ...
The simply connected group has trivial center and outer automorphism group a product of the non-zero real numbers and a group of order 2. Type VI 0 : This Lie algebra is the semidirect product of R 2 by R , with R where the matrix M has non-zero distinct real eigenvalues with zero sum.