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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 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).
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 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 ...
In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold.The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation.
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.
By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3).
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σ i, so that the Hermitian matrix is written as a Pauli vector. [2] In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as ...