Euler's rotation theorem

Matrix proof

A spatial rotation is a linear map in one-to-one correspondence with a 3 × 3 rotation matrix R that transforms a coordinate vector x into X, that is Rx = X. Therefore, another version of Euler's theorem is that for every rotation R, there is a nonzero vector n for which Rn = n; this is exactly the claim that n is an eigenvector of R associated with the eigenvalue 1. Hence it suffices to prove that 1 is an eigenvalue of R; the rotation axis of R will be the line μn, where n is the eigenvector with eigenvalue 1.

A rotation matrix has the fundamental property that its inverse is its transpose, that is

${\displaystyle {\mathbf{R}}^{\mathsf{T}}\mathbf{R}=\mathbf{R}{\mathbf{R}}^{\mathsf{T}}=\mathbf{I},}$

where I is the 3 × 3 identity matrix and superscript T indicates the transposed matrix.

Computing the determinant shows that a rotation matrix ⁠${\displaystyle \mathbf{R}}$⁠ has determinant ⁠${\displaystyle det(\mathbf{R})=\pm 1}$⁠. Indeed,

${\displaystyle \begin{array}{cc}1& =det(\mathbf{I})=det\left({\mathbf{R}}^{\mathsf{T}}\mathbf{R}\right)=det\left({\mathbf{R}}^{\mathsf{T}}\right)det(\mathbf{R})\\ & =det(\mathbf{R}{)}^2\end{array}}$

A rotation matrix with determinant +1 is a proper rotation, and one with a negative determinant −1 is an improper rotation, that is a reflection combined with a proper rotation.

It will now be shown that a proper rotation matrix R has at least one invariant vector n, i.e., Rn = n. Because this requires that (R − I)n = 0, we see that the vector n must be an eigenvector of the matrix R with eigenvalue λ = 1. Thus, this is equivalent to showing that det(R − I) = 0.

Use the two relations

${\displaystyle det(-\mathbf{A})=(-1{)}^{3}det(\mathbf{A})=-det(\mathbf{A})}$

for any 3 × 3 matrix A and

${\displaystyle det\left({\mathbf{R}}^{-1}\right)=1}$

(since det(R) = 1) to compute

${\displaystyle \begin{array}{cc}& det(\mathbf{R}-\mathbf{I})=det\left((\mathbf{R}-\mathbf{I}{)}^{\mathsf{T}}\right)\\ =& det\left({\mathbf{R}}^{\mathsf{T}}-\mathbf{I}\right)=det\left({\mathbf{R}}^{-1}-{\mathbf{R}}^{-1}\mathbf{R}\right)\\ =& det\left({\mathbf{R}}^{-1}(\mathbf{I}-\mathbf{R})\right)=det\left({\mathbf{R}}^{-1}\right)det(-(\mathbf{R}-\mathbf{I}))\\ =& -det(\mathbf{R}-\mathbf{I})\\ ⟹0=& det(\mathbf{R}-\mathbf{I}).\end{array}}$

This shows that λ = 1 is a root (solution) of the characteristic equation, that is,

${\displaystyle det(\mathbf{R}-\lambda \mathbf{I})=0\text{for}\lambda =1.}$

In other words, the matrix R − I is singular and has a non-zero kernel, that is, there is at least one non-zero vector, say n, for which

${\displaystyle (\mathbf{R}-\mathbf{I})\mathbf{n}=\mathbf{0}⟺\mathbf{R}\mathbf{n}=\mathbf{n}.}$

The line μn for real μ is invariant under R, i.e., μn is a rotation axis. This proves Euler's theorem.

A simpler proof can be obtained by noting that R has either one real eigenvalue -1 or 1 and two complex eigenvalues e^{\pm i\theta} or three real eigenvalues from {-1,1}. Since det(R)=1 being the product of its three eigenvalues, we obtain that either all of them are 1 or only one of them is 1. Otherwise, the eigenvalues are {-1,-1,-1} or {-1,1,1} or {-1, e^{\pm i\theta}} implying det(R)=-1 which contradicts det(R)=1. So, there is a vector n of unit length such that Rn = n.

I

Equivalence of an orthogonal matrix to a rotation matrix

Two matrices (representing linear maps) are said to be equivalent if there is a change of basis that makes one equal to the other. A proper orthogonal matrix is always equivalent (in this sense) to either the following matrix or to its vertical reflection:

${\displaystyle \mathbf{R}\sim \left(\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right),0\le \varphi \le 2\pi .}$

Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigenvalue, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation.

If R has more than one invariant vector then φ = 0 and R = I. Any vector is an invariant vector of I.

Excursion into matrix theory

In order to prove the previous equation some facts from matrix theory must be recalled.

An m × m matrix A has m orthogonal eigenvectors if and only if A is normal, that is, if A^†A = AA^†. This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:

${\displaystyle \mathbf{A}\mathbf{U}=\mathbf{U}\mathrm{diag}({\alpha }_1,\dots ,{\alpha }_m)⟺{\mathbf{U}}^{†}\mathbf{A}\mathbf{U}=\mathrm{diag}({\alpha }_1,\dots ,{\alpha }_m),}$

and U is unitary, that is,

${\displaystyle {\mathbf{U}}^{†}={\mathbf{U}}^{-1}.}$

The eigenvalues α₁, ..., α_m are roots of the characteristic equation. If the matrix A happens to be unitary (and note that unitary matrices are normal), then

${\displaystyle {\left({\mathbf{U}}^{†}\mathbf{A}\mathbf{U}\right)}^{†}=\mathrm{diag}\left({\alpha }_1^{\ast },\dots ,{\alpha }_m^{\ast }\right)={\mathbf{U}}^{†}{\mathbf{A}}^{-1}\mathbf{U}=\mathrm{diag}\left(\frac{1}{{\alpha }_1},\dots ,\frac{1}{{\alpha }_m}\right)}$

and it follows that the eigenvalues of a unitary matrix are on the unit circle in the complex plane:

${\displaystyle {\alpha }_k^{\ast }=\frac{1}{{\alpha }_k}⟺{\alpha }_k^{\ast }{\alpha }_k={\left|{\alpha }_k\right|}^2=1,k=1,\dots ,m.}$

Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its characteristic equation (an mth order polynomial in λ) has real coefficients, it follows that its roots appear in complex conjugate pairs, that is, if α is a root then so is α^∗. There are 3 roots, thus at least one of them must be purely real (+1 or −1).

After recollection of these general facts from matrix theory, we return to the rotation matrix R. It follows from its realness and orthogonality that we can find a U such that:

${\displaystyle \mathbf{R}\mathbf{U}=\mathbf{U}\left(\begin{array}{ccc}{e}^{i\varphi }& 0& 0\\ 0& e^{-i\varphi }& 0\\ 0& 0& \pm 1\end{array}\right)}$

If a matrix U can be found that gives the above form, and there is only one purely real component and it is −1, then we define ${\displaystyle \mathbf{R}}$ to be an improper rotation. Let us only consider the case, then, of matrices R that are proper rotations (the third eigenvalue is just 1). The third column of the 3 × 3 matrix U will then be equal to the invariant vector n. Writing u₁ and u₂ for the first two columns of U, this equation gives

${\displaystyle \mathbf{R}{\mathbf{u}}_1=e^{i\varphi }{\mathbf{u}}_1\text{and}\mathbf{R}{\mathbf{u}}_2=e^{-i\varphi }{\mathbf{u}}_2.}$

If u₁ has eigenvalue 1, then φ = 0 and u₂ has also eigenvalue 1, which implies that in that case R = I. In general, however, as ${\displaystyle (\mathbf{R}-e^{i\varphi }\mathbf{I}){\mathbf{u}}_1=0}$ implies that also ${\displaystyle (\mathbf{R}-e^{-i\varphi }\mathbf{I}){\mathbf{u}}_1^{\ast }=0}$ holds, so ${\displaystyle {\mathbf{u}}_2={\mathbf{u}}_1^{\ast }}$ can be chosen for ${\displaystyle {\mathbf{u}}_2}$. Similarly, ${\displaystyle (\mathbf{R}-\mathbf{I}){\mathbf{u}}_3=0}$ can result in a ${\displaystyle {\mathbf{u}}_3}$ with real entries only, for a proper rotation matrix ${\displaystyle \mathbf{R}}$. Finally, the matrix equation is transformed by means of a unitary matrix,

${\displaystyle \mathbf{R}\mathbf{U}\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{i}{\sqrt{2}}& 0\\ \frac{1}{\sqrt{2}}& \frac{-i}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right)=\mathbf{U}\underset{=\mathbf{I}}{\underbrace{\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{i}{\sqrt{2}}& 0\\ \frac{1}{\sqrt{2}}& \frac{-i}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ \frac{-i}{\sqrt{2}}& \frac{i}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right)}}\left(\begin{array}{ccc}{e}^{i\varphi }& 0& 0\\ 0& e^{-i\varphi }& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{i}{\sqrt{2}}& 0\\ \frac{1}{\sqrt{2}}& \frac{-i}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right)}$

which gives

${\displaystyle {{\mathbf{U}}^{\prime }}^{†}\mathbf{R}{\mathbf{U}}^{\prime }=\left(\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right)\text{ with }{\mathbf{U}}^{\prime }=\mathbf{U}\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{i}{\sqrt{2}}& 0\\ \frac{1}{\sqrt{2}}& \frac{-i}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right).}$

The columns of U′ are orthonormal as it is a unitary matrix with real-valued entries only, due to its definition above, that ${\displaystyle {\mathbf{u}}_1}$ is the complex conjugate of ${\displaystyle {\mathbf{u}}_2}$ and that ${\displaystyle {\mathbf{u}}_3}$ is a vector with real-valued components. The third column is still ${\displaystyle {\mathbf{u}}_3=}$ n, the other two columns of U′ are perpendicular to n. We can now see how our definition of improper rotation corresponds with the geometric interpretation: an improper rotation is a rotation around an axis (here, the axis corresponding to the third coordinate) and a reflection on a plane perpendicular to that axis. If we only restrict ourselves to matrices with determinant 1, we can thus see that they must be proper rotations. This result implies that any orthogonal matrix R corresponding to a proper rotation is equivalent to a rotation over an angle φ around an axis n.

Equivalence classes

The trace (sum of diagonal elements) of the real rotation matrix given above is 1 + 2 cos φ. Since a trace is invariant under an orthogonal matrix similarity transformation,

${\displaystyle \mathrm{T}\mathrm{r}\left[\mathbf{A}\mathbf{R}{\mathbf{A}}^{\mathsf{T}}\right]=\mathrm{T}\mathrm{r}\left[\mathbf{R}{\mathbf{A}}^{\mathsf{T}}\mathbf{A}\right]=\mathrm{T}\mathrm{r}[\mathbf{R}]\text{ with }{\mathbf{A}}^{\mathsf{T}}={\mathbf{A}}^{-1},}$

it follows that all matrices that are equivalent to R by such orthogonal matrix transformations have the same trace: the trace is a class function. This matrix transformation is clearly an equivalence relation, that is, all such equivalent matrices form an equivalence class.

In fact, all proper rotation 3 × 3 rotation matrices form a group, usually denoted by SO(3) (the special orthogonal group in 3 dimensions) and all matrices with the same trace form an equivalence class in this group. All elements of such an equivalence class share their rotation angle, but all rotations are around different axes. If n is an eigenvector of R with eigenvalue 1, then An is also an eigenvector of ARA^T, also with eigenvalue 1. Unless A = I, n and An are different.

Applications

Generators of rotations

Suppose we specify an axis of rotation by a unit vector [x, y, z], and suppose we have an infinitely small rotation of angle Δθ about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix ΔR is represented as:

${\displaystyle \mathrm{\Delta }R=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+\left[\begin{array}{ccc}0& z& -y\\ -z& 0& x\\ y& -x& 0\end{array}\right]\mathrm{\Delta }\theta =\mathbf{I}+\mathbf{A}\mathrm{\Delta }\theta .}$

A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ as ⁠θ/N⁠ where N is a large number, a rotation of θ about the axis may be represented as:

${\displaystyle R={\left(\mathbf{1}+\frac{\mathbf{A}\theta }{N}\right)}^N\approx e^{\mathbf{A}\theta }.}$

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 (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 (in Euclidean space) 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 G = ba^T − ab^T.

${\displaystyle \begin{array}{cc}\mathbf{x}& =\mathbf{a}\cos \alpha +\mathbf{b}\sin \alpha \\ \mathbf{y}& =-\mathbf{a}\sin \alpha +\mathbf{b}\cos \alpha \\ \cos \alpha & ={\mathbf{a}}^{\mathsf{T}}\mathbf{x}\\ \sin \alpha & ={\mathbf{b}}^{\mathsf{T}}\mathbf{x}\\ \mathbf{y}& =-{\mathbf{a}\mathbf{b}}^{\mathsf{T}}\mathbf{x}+{\mathbf{b}\mathbf{a}}^{\mathsf{T}}\mathbf{x}=\left({\mathbf{b}\mathbf{a}}^{\mathsf{T}}-{\mathbf{a}\mathbf{b}}^{\mathsf{T}}\right)\mathbf{x}\\ {\mathbf{x}}^{\prime }& =\mathbf{x}\cos \beta +\mathbf{y}\sin \beta \\ & =\left(\mathbf{I}\cos \beta +\left({\mathbf{b}\mathbf{a}}^{\mathsf{T}}-{\mathbf{a}\mathbf{b}}^{\mathsf{T}}\right)\sin \beta \right)\mathbf{x}\\ \mathbf{R}& =\mathbf{I}\cos \beta +\left({\mathbf{b}\mathbf{a}}^{\mathsf{T}}-{\mathbf{a}\mathbf{b}}^{\mathsf{T}}\right)\sin \beta \\ & =\mathbf{I}\cos \beta +\mathbf{G}\sin \beta \\ \mathbf{G}& ={\mathbf{b}\mathbf{a}}^{\mathsf{T}}-{\mathbf{a}\mathbf{b}}^{\mathsf{T}}\end{array}}$

To include vectors outside the plane in the rotation one needs to modify the above expression for R by including two projection operators that partition the space. This modified rotation matrix can be rewritten as an exponential function.

${\displaystyle \begin{array}{cc}{\mathbf{P}}_{\mathbf{a}\mathbf{b}}& =-{\mathbf{G}}^2\\ \mathbf{R}& =\mathbf{I}-{\mathbf{P}}_{\mathbf{a}\mathbf{b}}+\left(\mathbf{I}\cos \beta +\mathbf{G}\sin \beta \right){\mathbf{P}}_{\mathbf{a}\mathbf{b}}=e^{\mathbf{G}\beta }\end{array}}$

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.

Quaternions

It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of three independent numbers. Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called a quaternion.

While the quaternion described above does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary numbers.

Rotation calculation via quaternions has come to replace the use of direction cosines in aerospace applications through their reduction of the required calculations, and their ability to minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.

Generalizations

In higher dimensions, any rigid motion that preserves a point in dimension 2n or 2n + 1 is a composition of at most n rotations in orthogonal planes of rotation, though these planes need not be uniquely determined, and a rigid motion may fix multiple axes. Also, any rigid motion that preserves n linearly independent points, which span an n-dimensional body in dimension 2n or 2n + 1, is a single plane of rotation. To put it another way, if two rigid bodies, with identical geometry, share at least n points of 'identical' locations within themselves, the convex hull of which is n-dimensional, then a single planar rotation can bring one to cover the other accurately in dimension 2n or 2n + 1.

A rigid motion in three dimensions that does not necessarily fix a point is a "screw motion". This is because a composition of a rotation with a translation perpendicular to the axis is a rotation about a parallel axis, while composition with a translation parallel to the axis yields a screw motion; see screw axis. This gives rise to screw theory.