# Questions tagged [quaternions]

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

981 questions
16 views

### Does every order in a quaternion algebra have a nice integral basis?

Let $A$ be a quaternion algebra over a number field $k$ and let $O \subset A$ be an order. Does there exist always a basis $\{1,i,j,k\}$ of $O$ over the ring of integers $\mathcal{O}_k$ of $k$ such ...
10 views

53 views

### Is a quaternion a way to divide vectors? [on hold]

This is a naive question but I read popular science articles where it is stated that quaternions define vector division, without further explanations
53 views

### Left (right) eigenvalues of quaternion Matrix

I have problem with calculating left eigenvalues for quaternion Matrices. Let's take a look at article 'Geršgorin type theorems for quaternionic matrices - Fuzhen Zhang': Click The right eigenvalues ...
42 views

12 views

### How to move a point [x, y, z] in 3D space around a center [0,0,0], using quaternion readings from a sensor?

I've been browsing the web for far too long, and still can't find a solution to this issue for my student project. Mind you my math skills are not that great, so I'm desperate for any help I can get! ...
24 views

### Vector part of $q^* v r$, what does it mean?

It's not clear why the quaternions are closed under addition. All of the constructions I've seen make it clear why they're closed under multiplication, but not addition. Anyway, consider the ...
50 views

### Quaternion conjugation map is orthogonal linear transformation

The following concerns an exercise from an undergraduate level textbook on Lie groups, which provides an elementary proof of $SU(2)/\mathbb{Z}_2 \cong SO(3)$. Think of $Sp(1)$ as the group of unit-...
47 views

### Norm of Quaternion

Given a quaternion of the form, $$q= a + bi + cj + dk$$ Which is the norm of $q$? (1) $\sqrt{a^2+b^2+c^2+d^2}$ (2) $a^2+b^2+c^2+d^2$ This page from MathWorks says (1) but another page says (2). ...
46 views

### Finding quaternion, representing transformation from one vector to another [closed]

Intro. Previously, I've asked a question on converting rgb triple to quaternion. After that question I've managed to get unit quaternions. Since they were unit ...
14 views

### Euler angle in two given coordinates

I have two coordinates. One is standard and the other is arbitrary. A rotation is represented as ZYX in the standard coordinates. (yes, It is not exactly a Euler angle. I'm sorry about this) and ...
13 views

29 views

### If x,y are orthonormal vectors with quaternionic coefficients, can we say that $\bar{x},\bar{y}$ are orthonormal?

In Morton L. Curtis's book Matrix Groups, it is stated that the orthonormality of the rows of a quaternionic matrix implies the orthonormality of the columns (Chapter 2 Proposition 4). However, as ...
20 views

### Torque required to achieve a desired quaternion

I was hoping someone can either explain or direct me towards a source that can help me with the following problem (not for homework, more of a hobby). Given an object with a current quaternion $q_c$ ...
47 views

### Interpolation between three-dimensional rotations

I have to define a continuous function $g: [0, 1] \rightarrow \mathrm{SO}(3)$ such that $g(0) = I$ and $g(1) = R$ (a given rotation). I know we can do this kind of interpolation using quaternions ...
27 views

### Existence of conjugation on a quaternion algebra given a separable subalgebra

In Vignéras' book Arithmétique des algèbres de quaternions, a quaternion algebra $A$ over a field $K$ is defined as a $4$-dim central algebra for which there exists a separable $2$-dim (necessarily ...
256 views

758 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 ...
### Quaternion Group: Determine that $i^4 = 1$.
Suppose we are given the following presentation of the quaternion group: $Q_8 = \langle i, j, k \ | \ i^2 = j^2 = k^2 = ijk\rangle$ Is it obvious that $i^4 = 1$?
Let $A$ be an indefinite quaternion algebra (e.g. $(2,5)_\mathbb{Q}$), let $M$ be a maximal order in $A$ and let $\Gamma$ be the Fuchsian group derived from $M$. We will denote the group of units of \$...