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.
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 |
|---|---|
| $A^2 ∈ ℝ$ | $A^{-1} = \frac{A}{A^2}$ |
| $d = 3$ | $A^{-1} = \frac{ĀÂÃ}{AĀÂÃ}$ |
| $d = 4$ | $A^{-1} = \frac{B}{AB}, B = Ā[AĀ]_{3,4}$ |
| $d = 5$ | $A^{-1} = \frac{B}{AB}, B = ĀÂÃ[AĀÂÃ]_{1,4}$ |
Formulae for multivector square roots
| Special case | Formula |
|---|---|
| $A^2 ∈ ℝ, A^2 < 0$ | $\sqrt{A} = \frac{A + λ}{\sqrt{2λ}}, λ = \sqrt{-A^2}$ |
| $A^2 ∈ ℝ, A^2 > 0, I^2 = -1$ | $\sqrt{A} = \frac{A + Iλ}{(1 + I)\sqrt{λ}}, λ = \sqrt{A^2}$ |