Infinitesimal rotation matrix: Difference between revisions

Revision as of 05:29, 17 May 2023

Revision as of 05:33, 17 May 2023

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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==