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

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Is there a nice meaning to the geometric triple product?

Using geometric algebra, I can easily find the geometric tripleproduct of three vectors $a,b,c \in \mathbb{R}^3$ to be $$abc = a \left(b \cdot c \right) - b \left( c \cdot a \right) + c \left( a ...
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How are these definitions of the inertia tensor the same?

I'm looking for some help in understanding the inertia tensor (not the physics, just the math). I'm trying to figure out how to convert between the wedge product and tensor product definitions. ...
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Homotopic properties of the Spin group from geometric algebra

There are two possible ways to define the $\mathrm{Spin}(n)$ group of Euclidean $n$-space from $\mathrm{Pin}(n)$. First is that $\mathrm{Spin}(n)$ is the identity component of $\mathrm{Pin}(n)$. ...
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Clifford algebra and Spin group of 4-dimensional Euclidean space

I’m seeking for a straightforward construction of well-known $\mathrm{Spin}(4) = \mathrm{Spin}(3)\times\mathrm{Spin}(3)$ isomorphism using geometric algebra-based definition of “Spin”, without ...
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Applying geometric calculus to a matrix expression

Let $f(\mathbf x) = \mathbf x^T A \mathbf x + \mathbf b^T\mathbf x + c$, where $A$ is a square matrix, $\mathbf x, \mathbf b$ are column matrices and $c$ is a constant. Then the gradient of this ...
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About the definition of norm in Clifford algebra

I have seen two definitions for the norm in the Clifford algebra $\mathrm{Cℓ}_{p, q, r}$. According to one of them $\Vert x\Vert = ⟨x. x^\dagger⟩_0$, where the dagger stands for the reversal of the ...
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Struggling with connection between Clifford Algebra (/GA) and their matrix generators

As I thought I understood things, the Gamma matricies behave as the 4 orthogonal unit vectors of the Clifford algebra $\mathcal{Cl}_{1,3}(\mathbb C)$, (also the Pauli matricies are for the 3 of ...
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Squaring a vector using geometric algebra

I'm doing research involving clifford algebra and I'm having difficulty understanding this one axiom: $a^2 = g(a,a)$. It states that this is the square of a vector and dividing the original vector by ...
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(Geometric algebra) Acceleration of a particle with constant speed as a bivector-vector inner product

I've been working on (self-studying) Geometric Algebra for Physicists which, sadly, has no solutions manual. This is not a problem in general, but I feel like one of my solutions for a question asked ...
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119 views

Geometric algebra: Rotors

I've been (slowly) working my way through a book on geometric algebra and have found one part particularly confusing. I can understand the equation $e_1e_2=\exp(e_1e_2 \pi/2)$ Where the substiution ...
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From Orthogonal vectors to Useful Bivector

If we have set of orthogonal vectors (X) can we form a set of orthogonal bivectors from that set? I am trying to find if there is a way to get 'more information' from an orthogonal matrix by some ...
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Geometric Algebra/ Calculus for Physics

I don't know if this would be a better question for physics.SE, but I'll try here first: There is at least one good book on classical mechanics using the geometric algebra/ calculus (GA): New ...
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Why is the grade of the wedge product of two arbitrary blades the sum of the two blades' grades independently?

I'm reading Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry and it says that this is true of any two arbitrary blades. $\ grade( \textbf{ A} \wedge \textbf{B})=grade( ...
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Comparison of Hilbert space tensor product and wedge product

For Hilbert Spaces: $$(|0\rangle + |1\rangle)\otimes (|0\rangle + |1\rangle) = |00\rangle + |01\rangle + |10\rangle + |11\rangle.$$ where all results are column vectors \begin{eqnarray*} 0 ...
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Norm on a Geometric Algebra

In the literature, for example "New Foundations for Classical Mechanics" by David Hestenes, the author introduces a function on the Geometric Algebra $$||M||^2=\langle M M^\dagger \rangle_0,$$ where ...
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141 views

Geometric algebra and quantum field theory

How does the reformulation of QFT with GA look like? I read that GA can be applied to almost every kind of physics, but QFT is rarely mentioned. Is there a lot of research going on in this direction ...
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I would like to see a De Morgan's Law example from Geometric Algebra

thanks for your time and effort. I really appreciate your help. In the documents "www.cs.bham.ac.uk/~fauserb/pdf/theses/Habilschrift.pdf" page 25 by B. Fauser and "Grade Free Product Formulae from ...
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Product between multivectors

I just want to see if I understood this. Since the geometric product is associative and so on we can write for two multivectors A and B given by $A= ...
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what is vector $(\vec{a}\cdot \vec{b})\vec{c} + (\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$

Suppose we have three non orthogonal vectors in $R^3$ as $\vec{a}, \vec{b}, \vec{c}$. The vector of $(\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ is in the plane spanned by ...
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Proof that geometric product is associative

Geometric product has nice property since it is a ring and it is associative to multiplication, which is not the case for vector cross product. But besides it is an axiom for geometric product, in the ...
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126 views

Geometric Product

I have a problem with the geometric product: In my book the unit trivector is defined like this: $(e_{1}e_{2})e_{3}=e_{1}e_{2}e_{3}$ But that would mean $(e_{1}e_{2})e_{3}= (e_{1} \wedge e_{2})\cdot ...
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derivative formula $\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} \wedge\mathbf{r}) = (n-1)\mathbf{a}$

Assume $\mathbf{r}=\mathbf{x}−\mathbf{x}′$ is the position vector in $\mathbb{R}^n$, for constant $\mathbf{a}$, we have $$\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} ...
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derivative $\nabla \frac{\mathbf{r}}{r^k}$ in the context of Geometric Calculus

Suppose $\mathbf{r} = \mathbf{x - x'}$ is the position vector in $\mathbf{R^n}$, and $r = |\mathbf{r}| = |\mathbf{x - x'}|$. Do we have $\nabla \frac{\mathbf{r}}{r^k} = \frac{n-k-1}{r^k}$ or $\nabla ...
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A question on the dual relationship between the regressive product and the exterior product

I am trying to understand the following sentence, which I came across in a book: The underlying beauty of the Ausdehnungslehre is due to this symmetry [the duality between the regressive and ...
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Can every element of a geometric algebra be represented as the product of vectors?

An element of the n-dimensional geometric algebra $\Bbb G^n$ over the reals is of the form $M = \langle M \rangle + \langle M \rangle_1 + \langle M \rangle_2 + \cdots + \langle M \rangle_n$. My ...
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Projections in Geometric Algebra

Given a geometric algebra $G$. Let $I, A,B$ be an $n,r$ and $s$ blade, respectively. Let $P(*)=(*\rfloor I) I^{-1}$ be the projection operator onto $I$. Suppose that $P(A)=A$. The problem is to ...
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How to define integration over the boundary of a curve?

When learning about Stokes' theorem ($\int_{\partial \Omega} \omega=\int_{\Omega} \mathrm d \omega$), we are told that it is just a generalization of the 2nd Fundamental Theorem of Calculus $(\int_a^b ...
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Proof that every noninvertiable multivector has an idempotent factor.

Prove that every multivector which does not have an inverse has an idempotent for a factor. Define an idempotent as a multivector $A$ with the property that $A^2=A$ and $A \neq 1$. I can ...
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What good is the Commutator product?

In geometric algebra, the commutator product is defined as $A \times B = \frac 1 2 (AB - BA)$. From linear algebra, I remember that the commutator of matrices is $[A, B] = AB - BA$ and the commutator ...
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Question about domains of Geometric (Clifford) Calculus functions

For those of you familiar with Geometric Calculus, and in particular, the book Vector and Geometric Calculus by Alan MacDonald, maybe you can explain something to me. On pg 16, MacDonald defines the ...
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Proving det(A) property with outermorphism definition

Let A be a square matrix. Prove that exchanging two columns of A changes the sign of $\det(A)$. **Note: I'm pretty sure this is supposed to be "adjacent columns" Source: "Linear and Geometric ...
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How is rejection defined: Question from *Linear and Geometric Algebra*

The rejection $rej_B(a) = (a \wedge B)/B$ is a linear transformation and so has an outermorphism extension. Show that the extension cannot serve as a rejection operator on $G^n$. Hint: Use the ...
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Rank and Nullity of Projection of Multivectors onto k-Blades

Describe the image and kernel of the projection function & verify the rank-nullity theorem: The projection function is defined as $P_B(M) = {(M \cdot B)}\ /\ B$ where M is a multivector in ...
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Are subspaces also subalgebras in Geometric Algebra?

Is a subspace of GA(n) closed under the geometric product? Say we let a k-blade represent a subspace of GA(n), where k < n. Does that also represent a subalgebra of GA(n)? I can see that we ...
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Geometric Algebra Question

Show that in 3D any pair of bivectors A and B have a common factor u such that A = au and B = bu. (a, b, u vectors -- au and bu are the geometric product) The only thing I can think of is to try ...
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geometric product between nonorthogonal basis and its reciprocal basis

This is related to the question of wedge product between nonorthogonal basis and its reciprocal basis in geometric algebra. If {$e_i$} is a set of basis that are not necessarily orthogonal, and ...
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131 views

wedge product between nonorthogonal basis and its reciprocal basis in geometric algebra

in the context of geometric algebra, what's the wedge product between basis and reciprocal basis? say, if {$e_i$} is a set of basis that are not necessarily orthogonal, and {$e^i$} is the ...
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165 views

what's the relationship of tensor and multivector

what's the relationship of multivector in geometric algebra and tensor? Is tensor a subset of multivector?
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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 ...
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What *is* the working form of a 1-vector in geometric algebra?

Consider the geometric algebra definition of 0-vectors (scalar), 1-vectors (vector), and the inner product. Let $a$ and $b$ be 1-vectors. Then $a + b = c$, where $c$ is another 1-vector. Now, ...
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inner product of trivector and bivector in geometric algebra

Hestenes's "New Foundations for Classical Mechanics" book (page 47, 1.1c) sets a problem to show: $\begin{aligned}\left( \mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c} \right) \cdot B=\mathbf{a} ...
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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, ...
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Questions about the conformal model of GA

The vectors $a$,$b$, and $e$ are null vectors in the GA generated by $R^{n+1,1}$ such that $a.e=b.e=1$ where $e$ is the vector representing a point at infinity. We are told that $a$ and $b$ (and ...
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The interpretation of the inner product betweeen two vectors

I am accustomed to Euclidean vector spaces where $x.a$ (both $x$ and $a$ being in $R^2$, say) can be interpreted as the magnitude of the projection of $x$ onto $a$ (or vice versa). Recently I have ...
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Help me evaluate this geometric product

We are in the geometric algebra generated by the vector space $R^{1,1}$. Consider three vectors, $e_+, e_-, e$ where $e_+^2=1$, $ e_-^2=-1$, $e_+.e_-=0$ and $e=e_+ +e_-$. It is straightforward to ...
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Is the Reversion map in Geometric Algebra well-defined?

I am studying from the book "Geometric Algebra for Physicists: by Chris Doran and Anthony Lasenby." In the book they define a map $\dagger : \mathcal G \to \mathcal G$, where $\mathcal G$ is a ...
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Cross Product Component Values

When taking the cross product, the x component of the perpendicular vector is the (signed) area of the yz projection of the parallelogram spanned by the two vectors it's orthogonal to-right? And ...
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Differential operator on a manifold in Geometric Calculus

In the context of Geometric Calculus, as stated in book Clifford Algebra to Geometric Calculus (pag. 142), let $M$ be a differentiable vector manifold, $F$ be a field on $M$ and $a$ be a tangent ...
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Adjoint of a Linear Function Acting on Bivectors

I've been self studying Doran and Lasenby's Geometric Algebra for Physicists, and I'm getting stuck on a small derivation in section 4.4.3 about the adjoint of linear functions. It's not big enough to ...
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multiplication table for conformal geometric algebra?

I'm trying to find the full multiplication table for conformal geometric algebra (should be a 32 by 32 matrix). It does not seem to be available in explicit form anywhere on the web. Can anybody help ...