2

I am writing a general geometric algebra library (dealing exclusively with euclidean bases) for Clojure to gain a better understanding on the topic, but how to invert a general multivector continues to cause me trouble.

In the first answer to Calculating the inverse of a multivector the formula for the inverse of a general multivector $B$ is stated to be

$$ B^{-1}= \frac{B^\dagger}{BB^\dagger}. $$

If this is the case, then that indicates that $BB^\dagger$ must evaluate to a non-zero scalar for $B$ to be invertible, correct? Otherwise it appears as though the definition of an inverse must be recursive.

Expanding $BB^\dagger$ given $B = a + be_1 + ce_2 + de_1e_2$ in $G_2$ results in the equation

$$ BB^\dagger = a^2 + b^2 + c^2 + d^2 + 2(ab + cd)e_1 + 2(ac - bd)e_2. $$

Given the assumption that $BB^\dagger$ must be scalar for $B$ to have an inverse, it follows that $ ab = -cd, $ and $ ac = bd $.

Is it the case that the general multivector $B$ is only invertible when the above identities between the components hold? Or is there a more general way to derive the inverse of a multivector which works in all cases given a euclidean set of basis vectors?

Suskeyhose
  • 21
  • A counterexample to the claim that $B^{-1}$ exists iff $BB^†$ is a (nonzero) scalar is $B = e_1 + 2e_1 e_2$. This has inverse $B^{-1} = -\frac13 e_1 -\frac23 e_1 e_2$, but $BB^† = 5 - 4e_2$. In this case, however, $B^2 = -3$, so $B^{-1} = \frac{B}{B^2}$. In my own geometric algebra library, I check for these two cases (where $BB$ or $BB^†$ is a scalar). I’m not clear about the general case though. – Jollywatt Sep 11 '22 at 00:43
  • The formula you cited doesn't work in general. Try the other answers to that question (in particular, the one by Mike Earnest and the one by me). – Don Hatch Sep 27 '24 at 07:55

1 Answers1

1

Sorry, I don't know so much about this myself, but I (google) discovered a paper in the physics arxiv that deals with the topic explicitly and seems to contain a lot of other relevant references:

Inverse of multivector: Beyond p+q=5 threshold A. Acus, A. Dargys

Another set of recipes are described on the following web page but I'm not so confident of their accuracy:

https://www.euclideanspace.com/maths/algebra/clifford/algebra/functions/inverse/manfred.htm

(I would have added this as a comment, but don't have adequate reputation)

doobs
  • 11