Inverses, Roots and Logarithms

In general, finding the inverse $A^{-1}$, square root $\sqrt{A}$ or logarithm $\log A$ of a general multivector $A$ is difficult. However, for certain cases, explicit formulae exist.

Multivector inverses

Any multivector $A$ has either no inverse or exactly one inverse $A^{-1}$ such that $AA^{-1} = A^{-1}A = 1$.

Explicit formulae for multivector inverses

For any metric in up to five dimensions, explicit formulae exist for the inverse of a multivector $A$. The implementation used in GeometricAlgebra.jl is mainly based on [3] and is described here.

For a multivector $A ∈ Cl(ℝ^d, ·)$ with metric $·$ in $d$ dimensions, let:

$Ā$ be the Clifford conjugate
$Â$ be the involute
$Ã$ be the reverse
$[A]_K$ denote the negation of grades $k ∈ K$, i.e.,

\[[A]_K = \sum_{k=0}^d ⟨A⟩_k · \begin{cases} -1 & \text{if } k ∈ K \\ +1 & \text{otherwise} .\end{cases}\]

Special case	Formula
$A^2 ∈ ℝ$	$A^{-1} = \frac{A}{A^2}$
$d = 3$	$A^{-1} = \frac{ĀÂÃ}{AĀÂÃ}$
$d = 4$	$A^{-1} = \frac{B}{AB}, B = Ā[AĀ]_{3,4}$
$d = 5$	$A^{-1} = \frac{B}{AB}, B = ĀÂÃ[AĀÂÃ]_{1,4}$

Faddeev–LeVerrier inverse algorithm

The Faddeev–LeVerrier inverse algorithm may be used to find the inverse of an $n \times n$ matrix $A$ in exactly $n$ steps (with one matrix multiplication per step). This algorithm can also be used to invert elements in a geometric algebra by considering a linear representation of multivectors as $n \times n$ matrices. [4] This method is implemented in inv_flv_method.

Crucially, we do not actually need to find a linear representation to use this method: we only need to know that one exists for a given dimension $n$. Assume the dimension of the smallest representation is $\text{repsize}(A)$.

Algorithm pseudocode

given a multivector $A$
$n := \text{repsize}(A)$
$c_n := 1$
$N \leftarrow 1$
for $k \in (n - 1, n - 2, ..., 1, 0)$
$N \leftarrow N + c_{k + 1} I$
$c_k := \frac{n}{k - n} A \odot N$
return $A^{-1} = -N/c_0$

The inverse exists if and only if $c_0 \ne 0$.

If $A$ is a $d$-dimensional multivector, then:

\[\text{repsize}(A) = \begin{cases} 1 & A = ⟨A⟩_0 \\ 2 & \text{$A$ is a pseudoscalar or (pseudo)vector} \\ 2^{\lfloor d/2 \rfloor} & \text{$A$ is even} \\ 2^{\lceil d/2 \rceil} & \text{in general} \\ \end{cases}\]

In some cases, the algorithm works in fewer steps than given above, particularly for multivectors of particular grades. However, I have not figured out how to determine the true minimum number of steps. (For example, by inspection, inverting a $5$-dimensional $3$-vector requires only $n = 4$ steps, not $8$ as suggested by this prescription.) It's an interesting problem!

Formulae for multivector square roots

Special case	Formula
$A^2 ∈ ℝ, A^2 < 0$	$\sqrt{A} = \frac{A + λ}{\sqrt{2λ}}, λ = \sqrt{-A^2}$
$A^2 ∈ ℝ, A^2 > 0, I^2 = -1, AI = IA$	$\sqrt{A} = \frac{A + Iλ}{(1 + I)\sqrt{λ}}, λ = \sqrt{A^2}$