View source for Infinitesimal rotation matrix

You do not have permission to edit this page, for the following reasons:

This IP address has been blocked from editing Wikipedia.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

The IP address or range 54.172.0.0/16 has been blocked by ‪JJMC89‬ for the following reason(s):

The IP address that you are currently using has been blocked because it is believed to be a web host provider or colocation provider. To prevent abuse, web hosts and colocation providers may be blocked from editing Wikipedia.
You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a proxy or VPN.
We recommend that you attempt to use another connection to edit. For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in.
If you do not have any other way to edit Wikipedia, you will need to request an IP block exemption.

How to appeal if you are confident that your connection does not use a colocation provider's IP address:
If you are confident that you are not using a web host, you may appeal this block by adding the following text on your talk page: {{unblock|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. Place any further information here. ~~~~}}. You must fill in the blank with your IP address for this block to be investigated. Your IP address can be determined here. Alternatively, if you wish to keep your IP address private you can use the unblock ticket request system. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.

Administrators: The IP block exemption user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a CheckUser needs to take a look at the account. This can be requested most easily at SPI Quick Checkuser Requests. Unblocking an IP or IP range with this template is highly discouraged without at least contacting the blocking administrator.

This block will expire on 05:35, 2 June 2025. Your current IP address is 54.172.240.154.

Even when blocked, you will usually still be able to edit your user talk page, as well as email administrators and other editors.

For information on how to proceed, please read the FAQ for blocked users and the guideline on block appeals. The guide to appealing blocks may also be helpful.

Other useful links: Blocking policy · Help:I have been blocked
This IP address range has been globally blocked.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

This block affects editing on all Wikimedia wikis.
The IP address or range 54.172.0.0/16 has been globally blocked by ‪Jon Kolbert‬ for the following reason(s):

Open proxy/Webhost: See the help page if you are affected

This block will expire on 18:27, 12 November 2028. Your current IP address is 54.172.240.154.

Even while globally blocked, you will usually still be able to edit pages on Meta-Wiki.

If you believe you were blocked by mistake, you can find additional information and instructions in the No open proxies global policy. Otherwise, to discuss the block please post a request for review on Meta-Wiki. You could also send an email to the stewards VRT queue at stewards@wikimedia.org including all above details.

Other useful links: Global blocks · Help:I have been blocked

You can view and copy the source of this page:

An '''infinitesimal rotation matrix''' or '''differential rotation matrix''' is a [[matrix (mathematics)|matrix]] representing an [[infinitesimal|infinitely]] small [[rotation]].

While a [[rotation matrix]] is an [[orthogonal matrix]] <math>R^\mathsf{T} = R^{-1}</math> representing an element of <math>SO(n)</math> (the [[special orthogonal group]]), the [[differential (mathematics)|differential]] of a rotation is a [[skew-symmetric matrix]] <math>A^\mathsf{T} = -A</math> in the [[tangent space]] <math>\mathfrak{so}(n)</math> (the [[special orthogonal Lie algebra]]), which is not itself a rotation matrix.

An infinitesimal rotation matrix has the form

:<math> I + d\theta \, A,</math>

where <math>I</math> is the identity matrix, <math>d\theta</math> is vanishingly small, and <math>A \in \mathfrak{so}(n).</math>

For example, if <math>A = L_x,</math> representing an infinitesimal three-dimensional rotation about the {{mvar|x}}-axis, a basis element of <math>\mathfrak{so}(3),</math>

:<math> dL_{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}. </math>

The computation rules for infinitesimal rotation matrices 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''.

==Discussion==
An '''infinitesimal rotation matrix''' is a [[skew-symmetric matrix]] where:
* As any [[rotation matrix]] has a single real eigenvalue, which is equal to +1, the corresponding [[eigenvector]] defines the [[rotation axis]].
* Its module defines an infinitesimal [[angular displacement]].

The shape of the matrix is as follows:
<math display="block">
  A = \begin{pmatrix}
     1          & -d\phi_z(t) &  d\phi_y(t) \\
     d\phi_z(t) &  1          & -d\phi_x(t) \\
    -d\phi_y(t) &  d\phi_x(t) &  1 \\
  \end{pmatrix}
</math>

===Associated quantities===
Associated to an infinitesimal rotation matrix <math>A</math> is an ''[[infinitesimal rotation tensor]]'' <math>d\Phi(t) = A - I</math>:

<math display="block">
  d\Phi(t) = \begin{pmatrix}
     0          & -d\phi_z(t) &  d\phi_y(t) \\
     d\phi_z(t) &  0          & -d\phi_x(t) \\
    -d\phi_y(t) &  d\phi_x(t) &  0 \\
  \end{pmatrix}
</math>

Dividing it by the time difference yields the ''[[angular velocity tensor]]'':

:<math>
\Omega = \frac{d\Phi(t)}{dt} = \begin{pmatrix}
   0           & -\omega_z(t) &  \omega_y(t) \\
   \omega_z(t) &  0           & -\omega_x(t) \\
  -\omega_y(t) &  \omega_x(t) &  0           \\
\end{pmatrix}
</math>

==Order of rotations==

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> To understand what this means, consider

:<math> dA_{\mathbf{x}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}.</math>

First, test the orthogonality condition, {{math|1=''Q''<sup>T</sup>''Q'' = ''I''}}. The product is

:<math> dA_\mathbf{x}^\textsf{T} \, dA_\mathbf{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 + d\theta^2 & 0 \\ 0 & 0 & 1 + d\theta^2 \end{bmatrix},</math>

differing from an identity matrix by second-order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.

Next, examine the square of the matrix,

:<math> dA_{\mathbf{x}}^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 - d\theta^2 & -2d\theta \\ 0 & 2\,d\theta & 1 - d\theta^2 \end{bmatrix}.</math>

Again discarding second-order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation,

:<math>dA_\mathbf{y} = \begin{bmatrix} 1 & 0 & d\phi \\ 0 & 1 & 0 \\ -d\phi & 0 & 1 \end{bmatrix}.</math>

Compare the products {{math|''dA''<sub>'''x'''</sub>&nbsp;''dA''<sub>'''y'''</sub>}} to {{math|''dA''<sub>'''y'''</sub>&nbsp;''dA''<sub>'''x'''</sub>}},

:<math>\begin{align}
 dA_{\mathbf{x}}\,dA_{\mathbf{y}} &= \begin{bmatrix} 1 & 0 & d\phi \\ d\theta\,d\phi & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix} \\
 dA_{\mathbf{y}}\,dA_{\mathbf{x}} &= \begin{bmatrix} 1 & d\theta\,d\phi & d\phi \\ 0 & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix}. \\
\end{align}</math>

Since <math>d\theta \, d\phi</math> is second-order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is ''commutative''. In fact,

:<math> dA_{\mathbf{x}}\,dA_{\mathbf{y}} = dA_{\mathbf{y}}\,dA_{\mathbf{x}},</math>

again to first order. In other words, {{em|the order in which infinitesimal rotations are applied is irrelevant}}.

This useful fact makes, for example, derivation of rigid body rotation relatively simple. But one must always be careful to distinguish (the first-order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from Lie algebra elements. When contrasting the behavior of finite rotation matrices in the [[Baker–Campbell–Hausdorff formula]] above with that of infinitesimal rotation matrices, where all the commutator terms will be second-order infinitesimals, one finds a bona fide vector space. Technically, this dismissal of any second-order terms amounts to [[Group contraction]].

== Generators of rotations ==
{{Main|Rotation matrix|Rotation group SO(3)|Infinitesimal transformation}}

Suppose we specify an axis of rotation by a unit vector [''x'', ''y'', ''z''], and suppose we have an [[Infinitesimal rotation|infinitely small rotation]] of angle Δ''θ'' about that vector. Expanding the rotation matrix as an infinite addition, and taking the first-order approach, the rotation matrix Δ''R'' is represented as:
: <math>\Delta R =
  \begin{bmatrix}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1
  \end{bmatrix}
  +
  \begin{bmatrix}
     0 &  z & -y \\
    -z &  0 &  x \\
     y & -x &  0
  \end{bmatrix}\,\Delta \theta
  = I + A\,\Delta\theta.
</math>

A finite rotation through angle ''θ'' about this axis may be seen as a succession of small rotations about the same axis. Approximating Δ''θ'' as ''θ''/''N'', where ''N'' is a large number, a rotation of ''θ'' about the axis may be represented as:
: <math>R = \left(I + \frac{A\theta}{N}\right)^N \approx e^{A\theta}.</math>

It can be seen that Euler's theorem essentially states that ''all'' rotations may be represented in this form. The product ''Aθ'' is the "generator" of the particular rotation, being the vector {{nowrap|(''x'', ''y'', ''z'')}} associated with the matrix ''A''. This shows that the rotation matrix and the [[axis-angle]] format are related by the exponential function.

One can derive a simple expression for the generator ''G''. One starts with an arbitrary plane<ref>in Euclidean space</ref> defined by a pair of perpendicular unit vectors ''a'' and ''b''. In this plane one can choose an arbitrary vector ''x'' with perpendicular ''y''. One then solves for ''y'' in terms of ''x'' and substituting into an expression for a rotation in a plane yields the rotation matrix ''R'', which includes the generator {{nowrap|1=''G'' = ''ba''<sup>T</sup> − ''ab''<sup>T</sup>}}.

:<math>\begin{align}
   x &=  a \cos\left( \alpha \right) + b \sin\left( \alpha \right) \\ 
   y &= -a \sin\left( \alpha \right) + b \cos\left( \alpha \right) \\ 
   \cos\left( \alpha \right) &= a^\mathrm{T} x \\
   \sin\left( \alpha \right) &= b^\mathrm{T} x \\ 
   y &= -ab^\mathrm{T} x + ba^\mathrm{T} x = \left( ba^\mathrm{T} - ab^\mathrm{T} \right)x \\ 
 \\ 
  x' &= x \cos\left( \beta \right) + y \sin\left( \beta \right) \\ 
     &= \left[ I \cos\left( \beta \right) + \left( ba^\mathrm{T} - ab^\mathrm{T} \right) \sin\left( \beta \right) \right]x \\
 \\ 
  R &= I \cos\left( \beta \right) + \left( ba^\mathrm{T} - ab^\mathrm{T} \right) \sin\left( \beta \right) \\ 
    &= I \cos\left( \beta \right) + G \sin\left( \beta \right) \\ 
 \\ 
  G &= ba^\mathrm{T} - ab^\mathrm{T} \\ 
\end{align}</math>

To include vectors outside the plane in the rotation one needs to modify the above expression for ''R'' by including two [[Projection (linear algebra)|projection operators]] that partition the space. This modified rotation matrix can be rewritten as an [[Matrix exponential#Rotation case|exponential function]].

: <math>\begin{align}
 P_{ab} &= -G^2 \\ 
      R &= I - P_{ab} + \left[ I \cos\left( \beta \right) + G \sin\left( \beta \right) \right] P_{ab} = e^{G\beta} \\ 
\end{align}</math>

Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the [[Lie algebra]] of the rotation group.

== 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 {{math|''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.{{efn|Note that this exponential  map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to 3rd order, 
<math>e^{2A} - \frac{I+A}{I-A} = - \frac{2}{3} A^3 +\mathrm{O}  (A^4)  ~.  </math> <br />
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'')}}.}}

An important practical example is the {{math|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|'''su'''(2) ≅ '''R'''<sup>3</sup>}}, {{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.

Using [[Rodrigues' rotation formula#Matrix notation|Rodrigues' rotation formula on matrix form]] with {{math|1=''θ'' = {{frac|''θ''|2}} + {{frac|''θ''|2}}}}, together with standard [[List of trigonometric identities#Multiple-angle and half-angle formulae|double angle formulae]] one obtains,
: <math>\begin{align}
 \exp( A ) &{}= \exp(\theta(\boldsymbol{u\cdot L}))
           = \exp \left( \left[\begin{smallmatrix} 0 & -z \theta & y \theta \\ z \theta & 0&-x \theta \\ -y \theta & x \theta & 0 \end{smallmatrix}\right] \right)= \boldsymbol{I} + 2\cos\frac{\theta}{2}\sin\frac{\theta}{2}~\boldsymbol{u\cdot L} + 2\sin^2\frac{\theta}{2} ~(\boldsymbol{u\cdot L} )^2 ,
\end{align}</math>
This is the matrix for a rotation around axis {{math|'''u'''}}  by the angle {{mvar|θ}} in half-angle form. For full detail, see [[Rotation group SO(3)#Exponential map|exponential map SO(3)]].

Notice that for infinitesimal angles second-order terms can be ignored and remains {{math|1=exp(''A'') = ''I'' + ''A''}}

==Relationship to skew-symmetric matrices==

Skew-symmetric matrices over the field of real numbers form the [[tangent space]] to the real [[orthogonal group]] <math>O(n)</math> at the identity matrix; formally, the [[special orthogonal Lie algebra]]. In this sense, then, skew-symmetric matrices can be thought of as ''infinitesimal rotations''.

Another way of saying this is that the space of skew-symmetric matrices forms the [[Lie algebra]] <math>o(n)</math> of the [[Lie group]] <math>O(n).</math> The Lie bracket on this space is given by the [[commutator]]:

:<math>[A, B] = AB - BA.\,</math>

It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric:

: <math>\begin{align}
  {[}A, B{]}^\textsf{T} &= B^\textsf{T} A^\textsf{T} - A^\textsf{T} B^\textsf{T} \\
                        &= (-B)(-A) - (-A)(-B) = BA - AB = -[A, B] \, .
\end{align}</math>

The [[matrix exponential]] of a skew-symmetric matrix <math>A</math> is then an [[orthogonal matrix]] <math>R</math>:

:<math>R = \exp(A) = \sum_{n=0}^\infty \frac{A^n}{n!}.</math>

The image of the [[exponential map (Lie theory)|exponential map]] of a Lie algebra always lies in the [[Connected space|connected component]] of the Lie group that contains the identity element. In the case of the Lie group <math>O(n),</math> this connected component is the [[special orthogonal group]] <math>SO(n),</math> consisting of all orthogonal matrices with determinant 1. So <math>R = \exp(A)</math> will have determinant&nbsp;+1. Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that ''every'' orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix. In the particular important case of dimension <math>n=2,</math> the exponential representation for an orthogonal matrix reduces to  the well-known [[complex number#Polar form|polar form]] of a complex number of unit modulus. Indeed, if <math>n=2,</math> a special orthogonal matrix has the form
:<math>\begin{bmatrix}
  a &  -b \\
  b & \,a
\end{bmatrix},</math>

with <math>a^2 + b^2 = 1</math>. Therefore, putting <math>a = \cos\theta</math> and <math>b = \sin\theta,</math> it can be written
:<math>\begin{bmatrix}
    \cos\,\theta &  -\sin\,\theta \\
    \sin\,\theta & \,\cos\,\theta
  \end{bmatrix} = \exp\left(\theta\begin{bmatrix}
    0 &  -1 \\
    1 & \,0
  \end{bmatrix}\right),
</math>

which corresponds exactly to the polar form <math>\cos \theta + i \sin \theta = e^{i \theta}</math> of a complex number of unit modulus.

The exponential representation of an orthogonal matrix of order <math>n</math> can also be obtained starting from the fact that in dimension <math>n</math> any special orthogonal matrix <math>R</math>  can be written as <math>R = QSQ^\textsf{T},</math> where <math>Q</math> is orthogonal and S is a [[block matrix#Block diagonal matrix|block diagonal matrix]] with <math display="inline">\lfloor n/2\rfloor</math> blocks of order&nbsp;2, plus one of order 1 if <math>n</math> is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix&nbsp;''S'' writes as exponential of a skew-symmetric block matrix <math>\Sigma</math> of the form above, <math>S = \exp(\Sigma),</math> so that <math>R = Q\exp(\Sigma)Q^\textsf{T} = \exp(Q\Sigma Q^\textsf{T}),</math> exponential of the skew-symmetric matrix <math>Q\Sigma Q^\textsf{T}.</math> Conversely, the surjectivity of the exponential map, together with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices.

==See also==
*[[Euler's rotation theorem#Generators of rotations|Generators of rotations]]
*[[Rotation matrix#Infinitesimal rotations|Infinitesimal rotations]]
*[[Infinitesimal strain theory#Infinitesimal rotation tensor|Infinitesimal rotation tensor]]
*[[Infinitesimal transformation]]
*[[Rotation group SO(3)#Infinitesimal rotations]]

==Notes==
{{notelist}}

==References==
{{reflist}}

==Sources==
* {{Citation |last1=Goldstein |first1=Herbert |author1-link=Herbert Goldstein |author2-link=Charles P. Poole |last2=Poole |first2=Charles P. |last3=Safko |first3=John L. |year=2002<!-- January 15 --> |title=Classical Mechanics |edition=third |publisher=[[Addison Wesley]] |isbn=978-0-201-65702-9}}
* {{Citation
 | last=Wedderburn
 | first=Joseph H. M.
 | author-link=Joseph Wedderburn
 | year=1934
 | title=Lectures on Matrices
 | publisher=[[American Mathematical Society|AMS]]
 | isbn=978-0-8218-3204-2
 | url=https://scholar.google.co.uk/scholar?hl=en&lr=&q=author%3AWedderburn+intitle%3ALectures+on+Matrices&as_publication=&as_ylo=1934&as_yhi=1934&btnG=Search
 }}

[[Category:Rotation]]
[[Category:Mathematics of infinitesimals]]

Pages transcluded onto the current version of this page (help):

Template:Citation (view source) (protected)
Template:Efn (view source) (template editor protected)
Template:Em (view source) (template editor protected)
Template:Frac (view source) (template editor protected)
Template:Fraction (view source) (template editor protected)
Template:Fraction/styles.css (view source) (template editor protected)
Template:Harv (view source) (template editor protected)
Template:Harvard citation (view source) (template editor protected)
Template:Main (view source) (template editor protected)
Template:Main other (view source) (protected)
Template:Math (view source) (template editor protected)
Template:Mvar (view source) (template editor protected)
Template:Notelist (view source) (template editor protected)
Template:Nowrap (view source) (protected)
Template:Reflist (view source) (protected)
Template:Reflist/styles.css (view source) (protected)
Module:Arguments (view source) (protected)
Module:Check for unknown parameters (view source) (protected)
Module:Citation/CS1 (view source) (protected)
Module:Citation/CS1/COinS (view source) (protected)
Module:Citation/CS1/Configuration (view source) (protected)
Module:Citation/CS1/Date validation (view source) (protected)
Module:Citation/CS1/Identifiers (view source) (protected)
Module:Citation/CS1/Utilities (view source) (protected)
Module:Citation/CS1/Whitelist (view source) (protected)
Module:Citation/CS1/styles.css (view source) (protected)
Module:Footnotes (view source) (template editor protected)
Module:Footnotes/anchor id list (view source) (template editor protected)
Module:Footnotes/anchor id list/data (view source) (template editor protected)
Module:Footnotes/whitelist (view source) (template editor protected)
Module:Format link (view source) (template editor protected)
Module:Hatnote (view source) (template editor protected)
Module:Hatnote/styles.css (view source) (template editor protected)
Module:Hatnote list (view source) (template editor protected)
Module:Labelled list hatnote (view source) (template editor protected)
Module:Yesno (view source) (protected)

Return to Infinitesimal rotation matrix.

Retrieved from "https://en.wikipedia.org/wiki/Infinitesimal_rotation_matrix"