Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [geometric-algebras]

Geometric algebras are Clifford algebras over the real numbers. They are applied in geometry and theoretical physics.

Geometric algebras are Clifford algebras over the real numbers. They can be used as a tool to study vector algebra, and they can be applied to problems in geometry and theoretical physics.

385 questions
58
votes
10 answers

Good introductory book on geometric algebra

The title of the question already says it all but I would like to add that I would really like the book to be more about geometric algebra than its applications : it should contain theorems' proofs. Just adding that I have never taken a course on…
Toussaint
  • 51
30
votes
11 answers

Looking for a clear definition of the geometric product

In brief: I'm looking for a clearly-worded definition1 of the geometric product of two arbitrary multivectors in $\mathbb{G}^n$. I'm having a hard time getting my bearings in the world of "geometric algebra", even though I'm using as my guide an…
kjo
  • 14,904
27
votes
1 answer

What's the difference between geometric, exterior and multilinear algebra?

I've studied what I think is geometric algebra, but can't seem to understand the difference between it and exterior and multilinear algebra. And is it linked to Clifford and Grassmann algebras in any way?
Anivarya Kumar
  • 373
22
votes
5 answers

Is Geometric Algebra isomorphic to Tensor Algebra?

Is geometric algebra (Clifford algebra) isomorphic to tensor algebra? If so, how then would one relate a unique 2-vector (this is what I'm going to call a multivector that is the sum of a scalar, vector, and bivector) to every 2nd rank…
user137731
18
votes
1 answer

What is the difference between Projective Geometric, Clifford Algebra, Grassman Algebra, Geometric Algebra, Quaternion Algebra and Exterior Algebra?

Since a few years now, Special Interest Group on Computer Graphics have been shilling this new type of algebra that they advertise fixes all the problem with Linear Algebra like no Gimbal locks, error free transformations, co-ordinate free…
sigsegv
  • 593
18
votes
6 answers

Why can't rotations be represented by purely imaginary quaternions?

I imagine this question has a straightforward answer, but I haven't been able to think of it on my own. It's well-accepted that you can represent a rotation in three-space with a unit quaternion. In other words, any rotation can be represented by…
Draconis
  • 1,503
14
votes
3 answers

Why Is $\sqrt{\det(A^TA)}$ A Volume / Volume Factor?

The determinant of an $n\times n$ matrix is a volume / volume factor. So far, I'm good in my understanding. You take a linear map, encode it as a matrix, compute the volume of the parallelepiped (or whatever the proper name is) spanned by the column…
M.B.
  • 3,332
13
votes
5 answers

Intuition for geometric product being dot + wedge product

While I feel quite comfortable with the meaning of the dot and exterior products separately (parallelity and perpendicularity), I struggle to find meaning in the geometric product as the combination of the two given that one’s a scalar and the other…
Kevin Goodman
  • 131
12
votes
0 answers

Conflicting definitions of a spinor

I've come across two definitions of "spinors" that I'm having a hard time reconciling: Spinors are the "square root" of a null vector (see here, and also Cartan's book "The Theory of Spinors") Spinors are minimal ideals in a Clifford algebra (see…
eigenchris
  • 2,182
12
votes
4 answers

Why use geometric algebra and not differential forms?

This is somewhat similar to Are Clifford algebras and differential forms equivalent frameworks for differential geometry?, but I want to restrict discussion to $\mathbb{R}^n$, not arbitrary manifolds. Moreover, I am interested specifically in…
Chill2Macht
  • 22,055
  • 10
  • 67
  • 178
11
votes
5 answers

In Geometric Algebra, is there a geometric product between matrices?

Thanks for your help in advance. I literally just started to self-study about geometric algebra. I have some coursework background in linear algebra and was trying to make an educational bridge between what I know and what I'm trying to learn. My…
New-to-GA
  • 181
11
votes
3 answers

Are all manifolds in the usual sense also "vector manifolds"?

In geometric calculus, there is a concept of a vector manifold where the points are considered vectors in a general geometric algebra (a vector space with vector multiplication) which can then be shown to have the properties of a manifold (tangent…
Eric Schneider
  • 491
10
votes
1 answer

What are some simple examples I can use to demonstrate the power of geometric algebra?

What are some simple examples I can use to demonstrate the power of geometric algebra over "everyday" vector algebra? An alternative way of thinking of this question might be: what example demonstrates how geometric algebra simplifies a tedious…
bzm3r
  • 2,812
10
votes
7 answers

Calculating the inverse of a multivector

Given a multivector, what is the easiest way to compute its inverse? To take a concrete example, consider a bivector $ B = e_1(e_2 + e_3) $. To compute $ B^{-1} $, I can use the dual of $ B $: $$ B = e_1e_2e_3e_3 + e_1e_2e_2e_3 = I(e_3-e_2) = Ib…
user997712
  • 303
9
votes
1 answer

Basic multivector derivatives $∂_X X = n$ and $∂_X X^2 = 2X$, etc…

Using geometric algebra, one may define the multivector derivative $∂_X$ with respect to a general multivector $X$ as $$ ∂_X ≔ \sum_J ^J (_J * ∂_X) $$ where each “component” $_J * ∂_X$ is defined by $$ (_J * ∂_X)f(X) ≔…
Jollywatt
  • 668
1
2 3
…
25 26