Editing Rotation matrix (section)

== Group theory ==
Below follow some basic facts about the role of the collection of ''all'' rotation matrices of a fixed dimension (here mostly 3) in mathematics and particularly in physics where [[rotational symmetry]] is a ''requirement'' of every truly fundamental law (due to the assumption of '''isotropy of space'''), and where the same symmetry, when present, is a ''simplifying property'' of many problems of less fundamental nature. Examples abound in [[classical mechanics]] and [[quantum mechanics]]. Knowledge of the part of the solutions pertaining to this symmetry applies (with qualifications) to ''all'' such problems and it can be factored out of a specific problem at hand, thus reducing its complexity. A prime example – in mathematics and physics – would be the theory of [[spherical harmonics]]. Their role in the group theory of the rotation groups is that of being a [[representation space]] for the entire set of finite-dimensional [[irreducible representation]]s of the rotation group SO(3). For this topic, see [[Rotation group SO(3)#Spherical harmonics|Rotation group SO(3) § Spherical harmonics]].

The main articles listed in each subsection are referred to for more detail.

=== Lie group ===
{{main|Special orthogonal group|Rotation group SO(3)}}
The {{math|''n'' × ''n''}} rotation matrices for each {{mvar|n}} form a [[group (mathematics)|group]], the [[special orthogonal group]], {{math|SO(''n'')}}. This [[algebraic structure]] is coupled with a [[topological structure]] inherited from <math>\operatorname{GL}_n(\R)</math> in such a way that the operations of multiplication and taking the inverse are [[analytic function]]s of the matrix entries. Thus {{math|SO(''n'')}} is for each {{mvar|n}} a [[Lie group]]. It is [[compact space|compact]] and [[connected space|connected]], but not [[simply connected]]. It is also a [[semi-simple group]], in fact a [[simple group]] with the exception SO(4).<ref>{{Harvtxt|Baker|2003}}; {{Harvtxt|Fulton|Harris|1991}}</ref> The relevance of this is that all theorems and all machinery from the theory of [[analytic manifold]]s (analytic manifolds are in particular [[smooth manifold]]s) apply and the well-developed representation theory of compact semi-simple groups is ready for use.

=== Lie algebra ===
{{main|Rotation group SO(3)#Lie algebra}}
The Lie algebra {{math|'''so'''(''n'')}} of {{math|SO(''n'')}} is given by
:<math>\mathfrak{so}(n) = \mathfrak{o}(n) = \left\{X \in M_n(\mathbb{R}) \mid X = -X^\mathsf{T} \right\},</math>
and is the space of skew-symmetric matrices of dimension {{math|''n''}}, see [[classical group]], where {{math|'''o'''(''n'')}} is the Lie algebra of {{math|O(''n'')}}, the [[orthogonal group]]. For reference, the most common basis for {{math|'''so'''(3)}} is
:<math>
 L_{\mathbf{x}} = \begin{bmatrix}0&0&0\\0&0&-1\\0&1&0\end{bmatrix} , \quad
 L_{\mathbf{y}} = \begin{bmatrix}0&0&1\\0&0&0\\-1&0&0\end{bmatrix} , \quad
 L_{\mathbf{z}} = \begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}.
</math>

=== Exponential map ===
{{main|Rotation group SO(3)#Exponential map|Matrix exponential}}
Connecting the Lie algebra to the Lie group is the [[exponential map (Lie theory)|exponential map]], which is defined using the standard [[matrix exponential]]  series for {{mvar|e<sup>A</sup>}}<ref>{{Harv|Wedderburn|1934|loc=§8.02}}</ref> For any [[skew-symmetric matrix]] {{mvar|A}}, {{math|exp(''A'')}} is always a rotation matrix.<ref group="nb">Note that this exponential map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to the third order,
:<math>e^{2A} - \frac{I+A}{I-A}=- \tfrac{2}{3} A^3 +\mathrm{O} \left(A^4\right) . </math>
Conversely, a [[skew-symmetric matrix]] {{mvar|A}} specifying a rotation matrix through the Cayley map specifies the ''same'' rotation matrix through the map {{math|exp(2 artanh ''A'')}}.</ref>

An important practical example is the {{nowrap|3 × 3}} case. In [[rotation group SO(3)]], it is shown that one can identify every {{math|''A'' ∈ '''so'''(3)}} with an Euler  vector {{math|1='''ω''' = ''θ'''''u'''}}, where {{math|1='''u''' = (''x'', ''y'', ''z'')}} is a unit magnitude vector.

By the properties of the identification <math>\mathbf{su}(2) \cong \mathbb{R}^3</math>, {{math|'''u'''}} is in the null space of {{mvar|A}}. Thus, {{math|'''u'''}} is left invariant by {{math|exp(''A'')}} and is hence a rotation axis.

According to [[Rodrigues' rotation formula#Matrix notation|Rodrigues' rotation formula on matrix form]], one obtains,

:<math>\begin{align}
 \exp( A ) &= \exp\bigl(\theta(\mathbf{u}\cdot\mathbf{L})\bigr) \\
           &= \exp \left( \begin{bmatrix} 0 & -z \theta & y \theta \\ z \theta & 0&-x \theta \\ -y \theta & x \theta & 0 \end{bmatrix} \right) \\
           &= I + \sin \theta \ \mathbf{u}\cdot\mathbf{L} + (1-\cos \theta)(\mathbf{u}\cdot\mathbf{L} )^2 ,
\end{align}</math>

where
: <math> \mathbf{u}\cdot\mathbf{L} = \begin{bmatrix} 0 & -z & y \\ z & 0&-x \\ -y & x & 0 \end{bmatrix} .</math>

This is the matrix for a rotation around axis {{math|'''u'''}} by the angle {{mvar|θ}}. For full detail, see [[Rotation group SO(3)#Exponential map|exponential map SO(3)]].

=== Baker–Campbell–Hausdorff formula ===
{{main|Baker–Campbell–Hausdorff formula|Rotation group SO(3)#Baker–Campbell–Hausdorff formula}}
The BCH formula provides an explicit expression for {{math|1=''Z'' = log(''e''<sup>''X''</sup>''e''<sup>''Y''</sup>)}} in terms of a series expansion of nested commutators of {{mvar|X}} and {{mvar|Y}}.<ref>{{Harvnb|Hall|2004|loc=Ch.&nbsp;3}}; {{Harvnb|Varadarajan|1984|loc=§2.15}}</ref> This general expansion unfolds as<ref group=nb>For a detailed derivation, see [[Derivative of the exponential map]]. Issues of convergence of this series to the right element of the Lie algebra are here swept under the carpet. Convergence is guaranteed when {{math|{{norm|''X''}} + {{norm|''Y''}} < log 2}} and {{math|{{norm|''Z''}} < log 2}}. If these conditions are not fulfilled, the series may still converge. A solution always exists since {{math|exp}} is onto{{clarify|date=June 2017}} in the cases under consideration.</ref>
:<math> Z = C(X, Y) = X + Y + \tfrac{1}{2} [X, Y] + \tfrac{1}{12} \bigl[X,[X,Y]\bigr] - \tfrac{1}{12} \bigl[Y,[X,Y]\bigr] + \cdots .</math>

In the {{nowrap|3 × 3}} case, the general infinite expansion has a compact form,<ref>{{Harv|Engø|2001}}</ref>
:<math>Z = \alpha X + \beta Y + \gamma[X, Y],</math>
for suitable trigonometric function coefficients, detailed in the [[Rotation group SO(3)#Baker–Campbell–Hausdorff formula|Baker–Campbell–Hausdorff formula for SO(3)]].

As a group identity, the above holds for ''all faithful representations'', including the doublet (spinor representation), which is simpler. The same explicit formula thus follows straightforwardly through Pauli matrices; see the [[Pauli matrices#Exponential of a Pauli vector|{{nowrap|2 × 2}} derivation for SU(2)]]. For the general {{math|''n'' × ''n''}} case, one might use Ref.<ref>{{Cite journal | doi = 10.3842/SIGMA.2014.084|last1 = Curtright|last2 = Fairlie|last3 = Zachos |first1 = T L |first2 = D B |first3 = C K|author-link=Thomas Curtright|author-link2=David Fairlie|author-link3=Cosmas Zachos|year = 2014|title = A compact formula for rotations as spin matrix polynomials| journal =SIGMA| volume=10| page=084|arxiv = 1402.3541|bibcode = 2014SIGMA..10..084C|s2cid = 18776942}}</ref>

=== Spin group ===
{{main|Spin group|Rotation group SO(3)#Connection between SO(3) and SU(2)}}
The Lie group of {{math|''n'' × ''n''}} rotation matrices, {{math|SO(''n'')}}, is not [[simply connected space|simply connected]], so Lie theory tells us it is a homomorphic image of a [[universal covering group]]. Often the covering group, which in this case is called the [[spin group]] denoted by {{math|Spin(''n'')}}, is simpler and more natural to work with.<ref>{{Harvnb |Baker|2003|loc=Ch.&nbsp;5}}; {{Harvnb|Fulton|Harris|1991|pp=299–315}}</ref>

In the case of planar rotations, SO(2) is topologically a [[circle]], {{math|''S''<sup>1</sup>}}. Its universal covering group, Spin(2), is isomorphic to the [[real line]], {{math|'''R'''}}, under addition. Whenever angles of arbitrary magnitude are used one is taking advantage of the convenience of the universal cover. Every {{nowrap|2 × 2}} rotation matrix is produced by a countable infinity of angles, separated by integer multiples of 2{{pi}}. Correspondingly, the [[fundamental group]] of {{math|SO(2)}} is isomorphic to the integers, {{math|'''Z'''}}.

In the case of spatial rotations, [[Rotation group SO(3)|SO(3)]] is topologically equivalent to three-dimensional [[real projective space]], {{math|'''RP'''<sup>3</sup>}}. Its universal covering group, Spin(3), is isomorphic to the {{nowrap|3-sphere}}, {{math|''S''<sup>3</sup>}}. Every {{nowrap|3 × 3}} rotation matrix is produced by two opposite points on the sphere. Correspondingly, the [[fundamental group]] of SO(3) is isomorphic to the two-element group, {{math|'''Z'''<sub>2</sub>}}.

We can also describe Spin(3) as isomorphic to [[quaternion]]s of unit norm under multiplication, or to certain {{nowrap|4 × 4}} real matrices, or to {{nowrap|2 × 2}} complex [[special unitary group|special unitary matrices]], namely SU(2). The covering maps for the first and the last case are given by
:<math> \mathbb{H} \supset \{q \in \mathbb{H}: \|q\| = 1\} \ni w + \mathbf{i}x + \mathbf{j}y + \mathbf{k}z \mapsto
  \begin{bmatrix}
    1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\
    2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\
    2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2
  \end{bmatrix} \in \mathrm{SO}(3),
</math>
and
:<math>\mathrm{SU}(2) \ni \begin{bmatrix}
               \alpha &             \beta \\
    -\overline{\beta} & \overline{\alpha}
  \end{bmatrix} \mapsto 
  \begin{bmatrix}
    \frac{1}{2}\left(\alpha^2 - \beta^2 + \overline{\alpha^2} - \overline{\beta^2}\right) &
       \frac{i}{2}\left(-\alpha^2 - \beta^2 + \overline{\alpha^2} + \overline{\beta^2}\right) &
      -\alpha\beta - \overline{\alpha}\overline{\beta} \\
    \frac{i}{2}\left(\alpha^2 - \beta^2 - \overline{\alpha^2} + \overline{\beta^2}\right) &
       \frac{i}{2}\left(\alpha^2 + \beta^2 + \overline{\alpha^2} + \overline{\beta^2}\right) &
      -i\left(+\alpha\beta - \overline{\alpha}\overline{\beta}\right) \\
    \alpha\overline{\beta} + \overline{\alpha}\beta &
       i\left(-\alpha\overline{\beta} + \overline{\alpha}\beta\right) &
       \alpha\overline{\alpha} - \beta\overline{\beta}
  \end{bmatrix} \in \mathrm{SO}(3).
</math>

For a detailed account of the {{nowrap|SU(2)-covering}} and the quaternionic covering, see [[Rotation group SO(3)#Connection between SO(3) and SU(2)|spin group SO(3)]].

Many features of these cases are the same for higher dimensions. The coverings are all two-to-one, with {{math|SO(''n'')}}, {{math|''n'' > 2}}, having fundamental group {{math|'''Z'''<sub>2</sub>}}. The natural setting for these groups is within a [[Clifford algebra]]. One type of action of the rotations is produced by a kind of "sandwich", denoted by {{math|''qvq''<sup>∗</sup>}}. More importantly in applications to physics, the corresponding spin representation of the Lie algebra sits inside the Clifford algebra. It can be exponentiated in the usual way to give rise to a {{nowrap|2-valued}} representation, also known as [[projective representation]] of the rotation group. This is the case with SO(3) and SU(2), where the {{nowrap|2-valued}} representation can be viewed as an "inverse" of the covering map. By properties of covering maps, the inverse can be chosen ono-to-one as a local section, but not globally.

=== Infinitesimal rotations ===
{{main|Infinitesimal rotation matrix}}
The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or ''infinitesimal rotation matrix'' has the form
:<math> I + A \, d\theta ,</math>
where {{math|''dθ''}} is vanishingly small and {{math|''A'' ∈ '''so'''(n)}}, for instance with {{math|1=''A'' = ''L''<sub>''x''</sub>}},
:<math> dL_{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}. </math>

The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.<ref>{{Harv|Goldstein|Poole|Safko|2002|loc=§4.8}}</ref> It turns out that ''the order in which infinitesimal rotations are applied is irrelevant''. To see this exemplified, consult [[rotation group SO(3)#Infinitesimal rotations|infinitesimal rotations SO(3)]].