For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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1answer
31 views

Finding quaternion that transforms to particular basis

I want to find a quaternion $x \in{\mathbb{H}} $ that transforms (rotates) the $ i,j,k $ basis to a particular basis. In equations: $$ x i x^{-1} = a_1 $$ $$ x j x^{-1} = a_2 $$ $$ x k x^{-1} = a_3 ...
0
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1answer
40 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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0answers
12 views

Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this ...
0
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1answer
37 views

There is no group whose quotient by the center is isomorphic to the quaternion group [duplicate]

I have to prove that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$. Anytone can give me an idea to begin? thanks
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3answers
75 views

Are $i,j,k$ commutative?

I am trying to understand quaternions. I read that Hamilton came up with the great equation: A) $i^2 = j^2 = k^2 = ijk = −1$ In this equation I understand that $i,j,k$ are complex numbers. Later ...
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3answers
66 views

Conversion of rotation matrix to quaternion

We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & ...
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0answers
7 views

Find path between two attitudes subject to body rate constraints

Here's my problem. I have an initial orientation and angular velocity of a body and a final orientation and velocity occurring at a specified time in the future. I have control over how input ...
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0answers
29 views

Rotation plane on the sphere (quarternion)

I asked similar question on stackoverflow but still no answers.http://stackoverflow.com/questions/25185329/image-rotation-with-the-gyro-data-math I assume it is more math than programming problem. ...
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2answers
45 views

Derive a quaternion from three axis

My problem originates from some code that I'm writing to parse an obscure file-type in which a geometric entity is defined in it's own 'local space', and a rotation and translation are provided to ...
0
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0answers
11 views

Multilinear quaternion interpolation

I'm looking for literature to study more on multilinear quaternion interpolation. Looking for 'trilinear interpolation' and 'tricubic interpolation' on Google Scholar or arxiv doesn't yield much ...
2
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2answers
53 views

Solving for a Rotation Matrix Equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$

I would like to solve for an equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$ where $R_1$, $R_2$, and $R_\mathrm{x}$ are 3x3 rotation matrices, $R_1$ and $R_2$ are known, and $R_\mathrm{x}$ is unknown. ...
2
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1answer
54 views

Tensor products and isomorphic algebras

I found that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus\mathbb{C}$ and that $ \mathbb{H} \otimes_{ \mathbb{R}} \mathbb{C} \simeq M_2( \mathbb{C})$. Could anybody hint me how ...
0
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1answer
64 views

Converting quaternions to spherical angles

Consider a situation where a beam is shot at a cube C from an arbitrary position P. The cube detects the angle of incidence relative to its $ x $ axis. The cube can be rotated and moved, and the ...
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0answers
36 views

can quaternions be expressed in terms of tensor products?

QUESTIONS does this arithmetic check out? if so, is there a geometric interpretation? note: my aim was to try to find a very simple but non-trivial example which might help me begin to understand ...
2
votes
1answer
35 views

Need help with this exercise about real division algebra

I am trying to solve the following exrcise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. I ...
0
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0answers
21 views

Set of quaternions that anti-commute

I tried to solve another exercise and I would be grateful if someone could tell me if my answer is right. This is the exercise: Characterize the pairs $x,y \in \mathbb H$ such that $xy = -yx$. ...
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0answers
31 views

identifying $\mathbb H$$^n$ with $\mathbb C^{2n}$

Let $X \in M_n(\mathbb H)$ (Hermetian field). It is possible to make $\mathbb H$$^n$ into a $2n$-dimensional vector space over $\mathbb C$, for example, by embedding $\mathbb C$ into $\mathbb H$ as ...
3
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1answer
53 views

Commuting quaternions

I tried to solve the following exercise, please could somebody tell me if I did it right?: Prove that non-real elements $x,y \in \mathbb H$ commute if and only if their imaginary parts are ...
1
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1answer
54 views

Isomorphism of quaternions with a matrix ring over real numbers

Let $\mathcal A$ be the algebra over the real numbers consisting of matrices of the form $$\begin{pmatrix} z&w\\ - \bar{w}& \bar{z} \end{pmatrix} \ (z, w \in \mathbb C). $$ $\mathcal A$ is in ...
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0answers
17 views

Finding component Velocity relative to Velocity

I am trying to find the velocity of an object in a particular Direction based on the current rotation and velocity of that object. I will try and illustrate the example: I have a space craft ...
0
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1answer
18 views

Error performing multiplication of Quaternions

Alright I'm going to try one last time to explain my problem with quaternions and multiplication of two quaternions in specific. This time hopefully I'll get an explanation that makes sense. (I posted ...
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0answers
23 views

Quaternions $\leftrightarrow$ Matrix - Source Code

Can anybody help me with a easy source code for the transformation quaternions $\leftrightarrow$ matrix? Any language programming is ok, doesn't matter if it about C/C++ or java, or written in Matlab. ...
0
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2answers
38 views

Smooth transition between two quaternions?

I am describing the orientation of an object with quaternion $q$. Now I want to describe (animate) smooth transition between orientations of $q_1$ and $q_2$. I was thinking that quaternion $q = q_1 ...
2
votes
1answer
50 views

Is there a (hypercomplex) number system, in which addition is **not** commutative

Now, we all know and love the quaternions, in which multiplication is not commutative, and the fact that, in all Clifford algebras, exponentiation is not commutative. Having looked at the properties ...
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0answers
24 views

Is there a term for extending a finite magma by adding coefficients from fields?

For example, the Quaternion numbers at their base have the Cayley table: $ * = \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j \\ j & -k & -1 & i \\ k & j ...
3
votes
1answer
59 views

What is the meaning of quaternion interpolation?

Suppose I take the average between two quaternions, how does one see the meaning of the resulting rotation to make sure it is sensible, unlike interpolating Euler angles? I'm looking for an argument ...
0
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2answers
25 views

Rotation between two vectors as a function of time (one parameter rational motion design)

Given a time varying vector: $\mathbf{w}(t) = \mathbf{u} + t\mathbf{v}$ I would like to find a rotation matrix $\mathbf{R}(t)$ that rotates the positive x-axis $[1, 0, 0]^T$ onto the vector ...
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0answers
125 views

Riemann Zeta function, quaternions and physics

Disclaimer: This question is rather vague, and thus probably not suitable for Mathoverflow, so I prefer to ask it here. I'm sorry if it doesn't meet the standards of this site. Several years ago, I ...
3
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0answers
89 views

Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b ...
2
votes
2answers
116 views

What is a good geometric interpretation of quaternion multiplication?

I understand that the formula for quaternion multiplication of $q_1=(s_1,\vec{v_1})$ by $q_2=(s_2,\vec{v_2})$ $q_1q_2=(s_1s_2-\vec{v_1}\cdot\vec{v_2}, \vec{v_1} \times\vec{v_2} + \vec{v_1}s_2 + ...
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0answers
51 views

How to obtain relative rotation?

I have two rotations, each of which can be described as a roll, pitch, and yaw (in radians): $$ r_1 = (3.14159, 1.57080, 1.6) $$ $$ r_2 = (3.14159, 1.57080, 1.4) $$ I am interested in the relative ...
3
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1answer
39 views

Proof that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$

In the wikipedia atrticle (http://en.wikipedia.org/wiki/Octonion) it is stated that "one can show that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, ...
6
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4answers
144 views

What's the intuition for extending $\mathbb{C}$ to $\mathbb{H}$?

It seems to me that there is a clear, intuitive reason for extending the real number system to the complex number system. Namely, some polynomial equations that have no solutions in $\mathbb{R}$ ...
2
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0answers
22 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
6
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1answer
120 views

Do there exist equations that cannot be solved in $\mathbb{C}$, but can be solved in $\mathbb{H}$?

Excluding polynomials (whose solutions are covered by the Fundamental Theorem of Algebra), do there exist any univariable equations that cannot be solved in the complex numbers, but can be solved ...
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0answers
39 views

Prove that $Q$ is a group under quaternion multiplication

Consider the subset $Q$ of the quaternions defined by $$Q=\{1,-1,i,-i,j,-j,k,-k\}.$$ Show that $Q$ is a group under quaternion multiplication. I know to prove something's a group, you must show ...
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0answers
124 views

Transform a vector to global frame and ignore rotation about one axis or Full tilt compensated magnetometer

Good day everyone. I would like to lock the rotation about one specified axis. For example, let`s imagine that we have a quaternion which desribes the orientation of our rigid body relative to the ...
1
vote
1answer
37 views

Question about quaternionic conjugation

The quaternionic conjugation is defined by $$\begin{aligned}i &\mapsto -i\\j&\mapsto -j\\k&\mapsto -k\end{aligned}$$ But since $ij=k$, shouldn't we have that $k = ij \mapsto (-i)(-j) = ...
0
votes
1answer
62 views

Is there infinitely many “complex units”

As we know, $i$ = $\sqrt{-1}$, a simple complex unit. In complex space of two dimensions, you graph an axis of $a+bi$ where $i$ is your second dimension axis. Now, you also know, in three and four ...
2
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1answer
56 views

Quaternions, Lie Groups and Lie Algebras. Steps to realize a paper. [closed]

I have to realize a paper about quaternions and Lie Groups and Lie Algebras. How can I realize the links between quaternions and Lie Groups & Algebras. Which books do you recommend me? First, I ...
1
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1answer
45 views

Can different quaternions represent the same orientation?

For a current project I'm working on I have to use quaternions to represent the orientation of an object. The piece of code I'm working on now integrates rotation rates to the quaternion representing ...
3
votes
1answer
46 views

Are roots of unity in hypercomplex algebras well defined?

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the ...
0
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1answer
62 views

Clifford Algebra and Fano Plane [closed]

Thank you for reading, I'm a novice, not a mathematician by trade this question could seem very simple (or even perhaps obvious) to many of you here. I've not yet found examples of this on the web. ...
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2answers
72 views

Dual quaternion inverse

Is it true that for every dual quaternion $Q$ I can find it's inverse such that $QQ^{-1} = 1?$ Using the usual definition $Q^{-1}=\frac{Q^{*}}{||Q||^2}$ doesn't work for me, since the dual part ...
2
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0answers
36 views

Does Shor's algorithm work for noncommutitive or nonassociative algebras?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, at least according to the Wikipedia article I read. This means that it can be used to solve factoring or ...
0
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1answer
45 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
1
vote
1answer
86 views

Complex number with 3 dimensions [duplicate]

I was looking back on complex analysis and asked myself: ''Why is there no complex number in 3 dimensions ?''. To place this question let me define with what I mean with 3 dimensions in the following. ...
1
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1answer
26 views

Rotation orientations in n-dimensions

I'm doing a change of variables that involves doing simple rotations on the standard basis vectors in R^n, and I'm wondering what the standard orientations are in n dimensions are and why. For ...
2
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0answers
39 views

Is complex symplectic structure equivalent to a quaternionic structure?

Let $S$ be a vector space over $\mathbb{C}$ of dimension $\operatorname{dim} S =2n$, let me denote by $S_\mathbb{R}$ real form of $S$ i.e. vector space over $\mathbb{R}$ of dimension $4n$. Is it true ...
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0answers
47 views

Quarternions from MPU and circumference of circles

First I should mention that my math skills are super basic. I do not understand formulas but I do understand pseudo code, C, C++, and other programming languages. I've been working on a electronics ...