Questions tagged [quaternions]

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

The ring of quaternions is a four dimensional division algebra over the real numbers. They are usually denoted as $\Bbb H$ in honor of the discoverer, William Rowan Hamilton.

The construction of the quaternions was given by Hamilton as follows: take three symbols $i,j,k$ and define $i^2=j^2=k^2=ijk=-1$. As a result, $ij=k$, and $jk=i$ and $ki=j$. Furthermore, $ji=-k$ and $kj=-i$ and $ik=-j$, so $kji=1$.

A quaternion is a linear combination $q=\alpha+\beta i+\gamma j +\delta k$ where $\alpha, \beta,\gamma,\delta\in \Bbb R$. Multiplication between quaternions is carried out by using the distributive rule and the rules for $i,j$ and $k$.

The quaternions turn out to be a noncommutative division ring. In fact, $\Bbb R$ and $\Bbb C$ and $\Bbb H$ are the only associative finite dimensional division rings over $\Bbb R$. They are also the only normed division algebras over $\Bbb R$.

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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} & m_{21}&m_{22}\end{bmatrix}}_{3\times 3}\tag 1 $. How do I accurately…
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Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)

I am trying to solve the following exercise: 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 am aware of Frobenius' theorem that there are only…
learner
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The sixth number system

In high school, I learned there are 5 number systems, namely: Natural numbers ($\Bbb N$) Integers ($\Bbb Z$) Rational numbers ($\Bbb Q$) Real numbers ($\Bbb R$) Complex numbers ($\Bbb C$) I remember one time our teacher told us that there is…
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Combining rotation quaternions

If I combine 2 rotation quaternions by multiplying them, lets say one represents some rotation around x axis and other represents some rotation around some arbitrary axis. The order of rotation matters, so the order of the quaternion multiplication…
Dan Webster
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How to form a mental image of $\mathbf {ijk}=-1$ in quaternions?

$\mathbf {ijk}=-1$ is part of the famously stone-carved formula by Sir William Rowan Hamilton, allowing the multiplication of triplets. There is already an intuition question on the topic of quaternions, and a beautifully illustrated post on 3D…
Antoni Parellada
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Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions (for example, the use of half-angles for…
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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…
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Square roots of quaternions

In class we saw that $-1$ has infinitely many square roots in the ring of quaternions. Is it possible to compute the square roots of a given nonreal quaternion? What do they look like?
RubyBlue
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Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm j \pm k}2 \right\} \subseteq…
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Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
chubbycantorset
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Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the complex field, where we can see a number as a $2$…
PPP
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Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?

Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group $SL(2,\mathbb{H})$? What I don't understand is how to…
Start wearing purple
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Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want to just memorize the formulas for using rotation…
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Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, and that there is a way to generalize…
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Is there a relationship between the cross product and quaternion multiplication?

I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus). Motivation for the question: I've been introduced to some properties of $\varepsilon_{ijk}$, e.g.…
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