Invariant decomposition

Any bivector $F$ in the geometric algebra $\backslash mathbb\{R\}\_\{p,q,r\}$ of total dimension $n\; =\; p+q+r$ can be decomposed into $k\; =\; \backslash lfloor\; n\; /\; 2\; \backslash rfloor$ orthogonal commuting simple bivectors that satisfy

$$F\; =\; F\_1\; +\; F\_2\; \backslash ldots\; +\; F\_\{k\}.$$

Defining $\backslash lambda\_i\; :=\; F\_i^2\; \backslash in\; \backslash mathbb\{C\}$, their properties can be summarized as $F\_i\; F\_j\; =\; \backslash delta\_\{ij\}\; \backslash lambda\_i\; +\; F\_i\; \backslash wedge\; F\_j$ (no sum). The $F\_i$ are then found as solutions to the characteristic polynomial

0 = (F_1 - F_i) (F_2 - F_i) \cdots (F_k - F_i).

Defining

$$W\_\{m\}\; =\; \backslash frac\{1\}\{m!\}\backslash langle\; F^\{m\}\backslash rangle\_\{2\; m\}\; =\; \backslash frac\{1\}\{m!\}\backslash ,\; \backslash underbrace\{F\; \backslash wedge\; F\; \backslash wedge\; \backslash ldots\; \backslash wedge\; F\}\_\{m\backslash \; \backslash text\{times\}\}$$

and $r\; =\; \backslash lfloor\; k/2\; \backslash rfloor$, the solutions are given by

$$F\_i$$

= \begin{cases}

\dfrac{\lambda_i^{r} W_0 + \lambda_i^{r-1} W_2 + \ldots + W_k}{\lambda_i^{r-1} W_1 + \lambda_i^{r-2} W_3 + \ldots + W_{k-1}} \quad & k \text{ even}, \\[10mu]

\dfrac{\lambda_i^{r} W_1 + \lambda_i^{r-1} W_3 + \ldots + W_k}{\lambda_i^{r} W_0 + \lambda_i^{r-1} W_2 + \ldots + W_{k-1}} & k \text{ odd}.

\end{cases}

The values of $\backslash lambda\_i$ are subsequently found by squaring this expression and rearranging, which yields the polynomial

$$\backslash begin\{aligned\}$$

0 &= \sum_{m=0}^{k} \langle W_{m}^2 \rangle_0 (- \lambda_i)^{k-m} \\[5mu]

&= (F_1^2 - \lambda_i) (F_2^2 - \lambda_i) \cdots (F_k^2 - \lambda_i).

\end{aligned}

By allowing complex values for $\backslash lambda\_i$, the counter example of Marcel Riesz can in fact be solved. This closed form solution for the invariant decomposition is only valid for eigenvalues $\backslash lambda\_i$ with algebraic multiplicity of 1. For degenerate $\backslash lambda\_i$ the invariant decomposition still exists, but cannot be found using the closed form solution.

A $2k$-reflection $R\; \backslash in\; \backslash text\{Spin\}(p,q,r)$ can be written as $R\; =\; \backslash exp(F)$ where $F\; \backslash in\; \backslash mathfrak\{spin\}(p,q,r)$ is a bivector, and thus permits a factorization

R = e^F = e^{F_1} e^{F_2} \cdots e^{F_k}.

The invariant decomposition therefore gives a closed form formula for exponentials, since each $F\_i$ squares to a scalar and thus follows Euler's formula:

R_i = e^{F_i} =

{\cosh}\bigl(\sqrt{\lambda_i}\bigr)

+ \frac{{\sinh}\bigl(\sqrt{\lambda_i}\bigr)}{\sqrt{\lambda_i}} F_i.

Carefully evaluating the limit $\backslash lambda\_i\; \backslash to\; 0$ gives

$$R\_i\; =\; e^\{F\_i\}\; =\; 1\; +\; F\_i,$$

and thus translations are also included.

Given a $2k$-reflection $R\; \backslash in\; \backslash text\{Spin\}(p,q,r)$ we would like to find the factorization into $R\_i\; =\; \backslash exp(F\_i)$. Defining the simple bivector

t(F_i) := \frac{{\tanh}\bigl(\sqrt{\lambda_i}\bigr)}{\sqrt{\lambda_i}} F_i,

where $\backslash lambda\_i\; =\; F\_i^2$. These bivectors can be found directly using the above solution for bivectors by substituting

$$W\_m\; =\; \backslash langle\; R\; \backslash rangle\_\{2m\}\; \backslash big/\; \backslash langle\; R\; \backslash rangle\_0$$

where $\backslash langle\; R\; \backslash rangle\_\{2m\}$ selects the grade $2m$ part of $R$. After the bivectors $t(F\_i)$ have been found, $R\_i$ is found straightforwardly as

$$R\_i\; =\; \backslash frac\{1\; +\; t(F\_i)\}\{\backslash sqrt\{1\; -\; t(F\_i)^2\}\}.$$

After the decomposition of $R\; \backslash in\; \backslash text\{Spin\}(p,q,r)$ into $R\_i\; =\; \backslash exp(F\_i)$ has been found, the principal logarithm of each simple rotor is given by

$$F\_i\; =\; \backslash text\{Log\}(R\_i)\; =\; \backslash begin\{cases\}$$

\dfrac{\langle R_i \rangle_2 }{\textstyle \sqrt{\langle R_i \rangle\vphantom)_2^2}} \;\text{arccosh}(\langle R_i \rangle) \quad & \lambda_i^2 \neq 0, \\[5mu]

\langle R_i \rangle_2 & \lambda_i^2 = 0.

\end{cases}

and thus the logarithm of $R$ is given by

$$\backslash text\{Log\}(R)\; =\; \backslash sum\_\{i=1\}^k\; \backslash text\{Log\}(R\_i).$$

So far we have only considered elements of $\backslash text\{Spin\}(p,q,r)$, which are $2k$-reflections. To extend the invariant decomposition to a $(2k+1)$-reflections $P\; \backslash in\; \backslash text\{Pin\}(p,q,r)$, we use that the vector part $r\; =\; \backslash langle\; P\; \backslash rangle\_1$ is a reflection which already commutes with, and is orthogonal to, the $2k$-reflection $R\; =\; r^\{-1\}\; P\; =\; P\; r^\{-1\}$. The problem then reduces to finding the decomposition of $R$ using the method described above.

The bivectors $F\_i$ are invariants of the corresponding $R\; \backslash in\; \backslash text\{Spin\}(p,q,r)$ since they commute with it, and thus under group conjugation

$$R\; F\_i\; R^\{-1\}\; =\; F\_i.$$

Going back to the example of Chasles' theorem as given in the introduction, the screw motion in 3D leaves invariant the two lines $F\_1$ and $F\_2$, which correspond to the axis of rotation and the orthogonal axis of translation on the horizon. 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:{{Cite book |last=Riesz |first=Marcel |editor-first1=E. Folke |editor-first2=Pertti |editor-last1=Bolinder |editor-last2=Lounesto |date=1993 |title=Clifford Numbers and Spinors |url=https://doi.org/10.1007/978-94-017-1047-3 |language=en-gb |doi=10.1007/978-94-017-1047-3|isbn=978-90-481-4279-8 }}

Can any bivector $F$ be decomposed into the direct sum of mutually orthogonal simple bivectors?

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Category:Group theory