Orthogonal group

The orthogonal group is the group of orthogonal matrices with real coefficients. The group operation of the orthogonal group is the matrix multiplication. In the orthogonal group is a Lie group of dimension. Since the determinant of an orthogonal matrix can only assume the values ​​, divided into two disjoint subsets ( topologically: connected components )

• The rotation group of all rotations ( orthogonal matrices with determinant) and
• Of all three reflections ( orthogonal matrices with determinant).

The subgroup is called the special orthogonal group. In particular, as the group of all rotations about an axis extending through the origin axis in three dimensional space is of great importance in many applications, such as computer graphics or the physics.

• 2.1 Topological Properties
• 2.2 Operation of the SO (n ) on the unit sphere
• 2.3 The Lie algebra of SO (n )

• 3.1 Description by axis and angle
• 3.2 Description by Euler angles
• 3.3 Description of Quaternion
• 3.4 Universal superposition of the SO (3)
• 3.5 Topology of the SO (3)
• 3.6 Finite subgroups of SO (3)

Orthogonal maps and matrices from algebraic point of view

Coordinate -free description

Starting from a - dimensional Euclidean vector space with a scalar product defined manner: An endomorphism is called orthogonal if the dot product is replaced, ie, if for all

Applies. A linear mapping gets exactly then the scalar product if it is length- and angle-preserving. The set of all orthogonal self-images by means of the orthogonal group, written as.

With respect to an orthonormal basis of orthogonal endomorphisms are represented by orthogonal matrices. Tantamount to this is the following formulation: Provides you with the standard scalar product, so the picture is exactly then orthogonal if the matrix is orthogonal.

Diagonalizability unitary matrices

Every orthogonal matrix is also of course a unitary matrix with real coefficients. Thus, it corresponds to a unitary figure after the spectral theorem for finite-dimensional unitary space is diagonalizable as a unitary matrix. The diagonal elements occurring with are exactly the eigenvalues ​​of. However, these are necessary on the amount one ( cf. unitary matrix ). They can therefore be in shape for certain, write to the sequence unique angle. Since the matrix has only real coefficients, occur while the nonreal eigenvalues ​​in pairs to each other conjugate complex numbers on. In the real case is not diagonalizable in general, but can also specify here a decomposition into a two-dimensional invariant subspaces or.

Impact on orthogonal matrices

At any orthogonal matrix can be a rotation of the coordinate system can be found, so that the matrix is " almost diagonal " form:

All specified herein coefficients have the value. The encountered matrices describing two-dimensional rotations by the angles of the form

Every part of it to a pair of complex conjugate eigenvalues. This of course applies if the number of diagonal elements represent with value and the number of diagonal elements with value. Apparently, if and rotation when the geometric as well as algebraic multiplicity of the eigenvalue, is an even number.

Plane rotation-reflection

In addition to the planar rotations that correspond to the matrices are also the rotation reflections

Orthogonal matrices. The eigenvalues ​​of are and; hence it is one of the mirror which, after a rotation of the coordinate system can be written to as.

Spatial rotation

After the normal form described above can be any rotation in space by choosing a suitable orthonormal basis by a matrix

Describe being with all special cases are detected. Said matrix describing a rotation about the axis. In particular, each real spatial rotation has an axis of rotation. Fischer illustrates this with the example of a soccer ball on the kick-off point: after the first goal, there are two opposite points on the ball, which are now exactly the same aligned to the stadium, as at the beginning of the game. The angle is uniquely determined due to the orientation-preserving nature of the authorized transformation matrices; this is accompanied by the experience known from everyday life that it is - at least theoretically - always, it is clear in which direction you have to turn a screw to tighten this fixed.

Spatial rotation-reflection

After the normal form described above can be any rotation-reflection in space by choosing a suitable orthonormal basis by a matrix

Describe being with all special cases are detected. Again, the angle is unique, unless you reverse the orientation of the room.

A double rotation in four-dimensional space

In four-dimensional space simultaneous rotation with two independent angles of rotation is possible:

One exchanges in a two-dimensional rotation, the two basis vectors, we obtain the rotation. This is not surprising, but at the same time you have changed the orientation of the plane. One exchanges now in the present example, at the same time the first with the second and also the third with the fourth base vector, the orientation is preserved, but will be out.

The orthogonal group as a Lie group

Based on the linear space of all matrices leads to the submanifold by the requirement that the matrix is ​​orthogonal, that is true. Orthogonal matrices are invertible because particular, is a subgroup of the general linear group.

Topological properties

As the general linear group is also the orthogonal group of two related components: matrices with positive or negative determinant in the case of the real; and the amount of orthogonal matrices with a force in the case of. Serge Lang gives an elegant proof of the Wegzusammenhang: you connect the unit matrix with a given rotation by a path within the. If one applies to every point on the way now the Gram -Schmidt orthogonalization to, we obtain a path that lies entirely in the. Since the multiplication of the diagonal matrix with its complement, in which supplies of a diffeomorphism, and the latter is connected.

Furthermore, as naturally compact. It is a closed subset of the unit sphere in.

Operation of the SO (n ) on the unit sphere

That operates in a natural way on the ( " matrix by vector" as described above). Since orthogonal maps are isometrically, the orbits of this operation are exactly the spheres around the origin. Thus, the operation restricts to a transitive operation on the unit sphere. The corresponding isotropy group of the canonical unit vector of the standard basis consists exactly of the, regarded as a subgroup of a. Matrix at the position One thus obtains the short exact sequence

Or the principal fiber bundle ( see also fiber bundle)

From this we conclude inductively that the fundamental group is isomorphic to for. It is thus similar to " twisted " as the Möbius strip. The fundamental group of the circle group ( see also number of turns ), since the topologically equivalent to the unit circle.

The Lie algebra of SO (n )

The Lie algebra so the tangent space of the point is to the unit matrix exactly from the skew-symmetric matrices. So is skew-symmetric, then the exponential map for matrices provides the associated Einparametergruppe

Obviously, a skew-symmetric matrix by the entries above the main diagonal is uniquely determined. Thus, the dimension is likewise clarified.

In the case of the associated Lie algebra is isomorphic to the cross product as the Lie bracket. To prove you have only the commutator of two generic, so educated, with three free variables, skew-symmetric matrices to calculate and compare the result with the formula for the cross product.

Three-dimensional rotations

Description by axis and angle

A rotation in three-dimensional space can be described by specifying an axis of rotation, ie, a vector of length one on the unit sphere, and a rotation angle. In the special case we obtain the identity map; for other angles, even in the case of a straight line with mirroring, the axis is clearly defined. By changing the orientation of the rotation axis can be a rotation also be regarded as a rotation angle.

The corresponding rotation can be explicitly specified by an associated rotation matrix ( see below).

Description by Euler angles

On Leonhard Euler is another description of rotations about three angles, the so-called Euler angles, back. Based on the standard basis Euler shows that each rotation as

Write leaves.

The three angle with the restriction to on singular spaces are uniquely determined: for naturally enough about one of the other two angles completely.

Euler angles are often used in physics; the description of the orbits of planets or asteroids by the so-called orbital elements due to this.

Description Quaternion

The Hamiltonian quaternions allow a very elegant description of spatial rotations. The quaternions form a four-dimensional algebra over the real numbers. As a basis we used four special quaternions, namely and. This is ( the multiplication is not commutative so ) and apply the following of William Rowan Hamilton specified calculation rules. With this multiplication rule - different, mathematically precise constructions of quaternion algebra can be found here - even to a skew field: For any nonzero quaternion can be an inverse quaternion calculate applies.

A quaternion is called pure if it can be described as a linear combination of three basis vectors and write. By virtue of the linear embedding

With, and we identify with the pure quaternions. Now can the multiplication rule for quaternions interpret geometrically: The product of two pure quaternions and is not back in, but it is

The pure component of the product of two pure quaternions corresponds to their vector product, while the scalar Share ( the coefficient in front of the base vector ) is their dot product.

To describe now the rotation by a quaternion, we first need an angle whose double, corresponding to the given. In addition, this also makes. We now consider the quaternion

This quaternion has unit length is ( with respect to the Standardskalarprodukts in ) and its inverse

Referring now any pure quaternion, it can be easy to show that the conjugate quaternion with

Again a pure quaternion (which they can be displayed only when exactly one ). It is now considered

In other words: the conjugation with the effect on pure quaternions as the rotation.

Universal covering of SO (3)

The above observed ambiguity in the choice of goes hand in hand with the two possible vectors for describing the axis: A specific rotation can be described exactly by two mutually inverse Einheitsquaternionen. Topologically clean it is in the amount of Einheitsquaternionen appeared to be a three-dimensional unit sphere in four-dimensional space. The quaternion multiplication gives it a Lie group structure. As such it is isomorphic to the special unitary group. As discussed in the previous section, provides the conjugation with a unit quaternion rotation. Apparently, these are a surjective group homomorphism, which is in a sufficiently small neighborhood of a diffeomorphism on its image. In other words, the picture

Is a two-sheeted overlay. There is simply connected, it is the universal covering of.

To understand the physical meaning of this universal cover, we return to the already considered above football. By appropriate markings on the ball can in principle be at any time determine the rotation that has taken place since the initiation of the ball. This results in a continuous path by starting at the unit matrix. Describing the identity matrix as by the identity element of ( alternatively you could use the antipodal opposite element in so ), so can now the whole way in a continuous manner lifts to a path through the. Even if you hit the ball to start the second half, the markings oriented according to exactly the same position again on the kick-off point (so that the path ends through the back at the point of the unit matrix ), it is not guaranteed that even the revamped way back in the one- quaternion is reached. With a probability of more in the latter ends; then you would have again rotated 360 ° with respect to an arbitrarily chosen axis to let it end also revamped the way to its starting point the ball. Since there is no superposition of the höherblättrige, it is not possible to detect the general rotations in space in a consistent manner even more finely.

Remarkably, used in quantum mechanics and not as a state space to describe the spin of a particle.

Topology of SO (3)

Each fiber of the recently described overlay

Consists of two antipodes points ( according to the two options for the choice of using ) the. Consequently, the homeomorphic to the quotient of opposite at identifying points. However, this gives exactly the three-dimensional real projective space.

Finite subgroups of SO (3)

The finite subgroups of are closely associated with spatial objects, which have a finite number of symmetries. Since any rotation about a cube the corresponding subgroup is conjugated with just this rotation in space, we are interested only for the conjugacy classes of finite subgroups of. These are:

• The cyclic groups, generated by a rotation of the -th part of a full- angle; for they are part of the full rotation group of a right prism over a regular -gon.
• The full rotation group of such a prism. In addition to that already described it includes also those rotations that act on the corner such as a regular reflection axis, but be simultaneously mirrors on the support plane of the pentagon to rotations of the room. These are the dihedral groups of order ( for formal, geometric interpretation for ).
• The full rotation group of a regular tetrahedron. It is isomorphic to the alternating group of order.
• The full rotation group of a regular octahedron, the so-called octahedral group of order. She is responsible for the dual octahedron cube simultaneously. It is isomorphic to the symmetric group.
• The full rotation group of a regular icosahedron (see icosahedral group ) of order. It also describes the rotations of the dodecahedron and is isomorphic to the alternating group.