# Tagged Questions

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

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### Using Padé approximants for the quaternion exponential

On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
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### Is there a residue theorem for Quaternions?

One of Complex Analysis's biggest contributions is the residue theorem. Is there a similar theorem in the field of Quaternion Analysis? (A glance at Wikipedia didn't pull anything that caught my eye)...
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### Are quaternions used in tensor analysis?

At the moment I am learning tensor analysis. In the books I study (civil engineering/mechanical books on tensors) there is a lot written about rotation matrices/tensors, however quaternions are not ...
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### Factoring quaternion into three parts

I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors. What I would like to know is angles (...
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### Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
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### Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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### 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 paper,...
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### 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 $f:\mathbb{H} \rightarrow \mathbb{H}$?
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### 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 ...
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### Why are quaternions (over the field of complex numbers) two-rowed matrix algebras?

I've enjoyed D.E. Littlewood's Skeleton Key to Mathematics up to chapter 8. After chapter 8, increasingly more statements are left without proof and I'm lost. He writes [...]it can be shown ...
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### Unwind quaternion multiplication

I am trying to understand quaterions division. Imagine I have the following equation, where every member is a quaternion: $$Q = (qq_1)(qq_2)...(qq_n)$$ I suppose that, if I maintain the order of ...
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### Slerp formula interpretation

I have a problem about spherical linear interpolation, or slerp for short. As linked, Wikipedia gives the following formula for an interpolation between quaternions ...
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### Grothendieck group of a quaternion algebra

Let $\mathcal{O}$ be a maximal order in a quaternion algebra over a number field. Then there is a notion of similarity of (left) $\mathcal{O}$-modules, and similarity classes. The set $S$ of such ...
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### 3D rotation by a complex angle around a complex axis.

Context In electromagnetism, it is common to meet complex angles. Snell's law can produce them (with metals and/or total internal reflection), and when we plug them into Fresnel's equations, the ...
Given Data and Specifications in Question If $q(t)$ represents the position vector as result of rotation with an angular velocity $\omega(t)$ in quaternions, then you can make the relationship \$...