It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups.
oriented reflections, the invariant decomposition theorem readsEvery
A well known special case is the Chasles' theorem, which states that any rigid body motion in
can be decomposed into a rotation around, followed or preceded by a translation along, a single line.
Both the rotation and the translation leave two lines invariant: the axis of rotation and the orthogonal axis of translation.
Since both rotations and translations are bireflections, a more abstract statement of the theorem reads "Every quadreflection can be decomposed into commuting bireflections".
In this form the statement is also valid for e.g. the spacetime algebra
, where any Lorentz transformation can be decomposed into a commuting rotation and boost.
orthogonal commuting simple bivectors that satisfy
are then found as solutions to the characteristic polynomial
are subsequently found by squaring this expression and rearranging, which yields the polynomial
, the counter example of Marcel Riesz can in fact be solved.
[1] This closed form solution for the invariant decomposition is only valid for eigenvalues
the invariant decomposition still exists, but cannot be found using the closed form solution.
The invariant decomposition therefore gives a closed form formula for exponentials, since each
squares to a scalar and thus follows Euler's formula:
has been found, the principal logarithm of each simple rotor is given by
Going back to the example of Chasles' theorem as given in the introduction, the screw motion in 3D leaves invariant the two lines
While the entire space undergoes a screw motion, these two axes remain unchanged by it.
The invariant decomposition finds its roots in a statement made by Marcel Riesz about bivectors:[2]Can any bivector
dimensional geometric algebra, it should be possible to find a maximum of
A first more general solution to the conjecture in geometric algebras
was given by David Hestenes and Garret Sobczyck.
[3] However, this solution was limited to purely Euclidean spaces.
(3DCGA) was published by Leo Dorst and Robert Jan Valkenburg, and was the first solution in a Lorentzian signature.
[4] Also in 2011, Charles Gunn was the first to give a solution in the degenerate metric
[5] This offered a first glimpse that the principle might be metric independent.
Then, in 2021, the full metric and dimension independent closed form solution was given by Martin Roelfs in his PhD thesis.
groups into orthogonal commuting factors which each follow Euler's formula, and to present closed form exponential and logarithmic functions for these groups.
Subsequently, in a paper by Martin Roelfs and Steven De Keninck the invariant decomposition was extended to include elements of