Questions tagged [quaternions]
For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers. Also for questions about quaternion algebras.
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Confusion in Partial Derivation of an Equation containing Quaternion
I found a way to rotate a 3D vector using a given unit quaternion. Thanks to this answer.
Now, let's say I want to rotate a gravity vector: $\overrightarrow{g} = \begin{bmatrix} g_x\\ g_y\\ g_z\\ \end{...
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Constructing Cyclic Division Algebras
I'm studying the construction of cyclic division algebras but I don't see how a given example holds.
According to the literature, we start with a finite extension $L$ of a number field $K$ such that ...
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Can the gamma function be generalized to quaternions and how? [duplicate]
The gamma function is a generalization of the operator !n. The question is: Can the concept of the gamma function be generalized to quaternion analysis and the use of quaternions, and how?
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Introduction to the Binary Tetrahedral group and the 24-cell
Context and introduction
I was playing with complex number sequences $Z_n=r_n\omega^n=u_n+iv_n$ represented in space and realized that it's always possible to associate up to 48 naturally symmetric ...
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How to find the angle required to rotate a parent frame such that one of the axes of its child frame points to a particular direction? [closed]
My problem is exactly similar to the one discussed here.
I want to calculate the the angle required to rotate a parent frame such that a particular axis of the child frame points in the direction of a ...
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geometry of two planes in $\mathbb{R}^4$
Problem: classify pairs of trivially intersecting 2-dimensional subspaces of $\mathbb{R}^4$, up to orthogonal transformations.
Linear algebra solution: if $(U, V)$ is such a pair, consider the ...
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Prove the binary icosahedral group is isomorphic to ${\rm SL}(2,5)$
I am having difficulty proving that the binary icosahedral group $2I$ is isomorphic to ${\rm SL}(2,5)$.
The binary icosahedral group $2I$ is a finite subgroup of $H^1$, where $H^1=\{q\in\mathbb{H}\mid ...
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Structure of automorphism groups of positive ternary quadratic forms
Let $f(x,y,z)=ax^2+by^2+cz^2+2ryz+2sxz+2txy$ be a positive definite integral ternary quadratic form (in Gauss's or Dickson's sense). Let $A(f)$ be its coefficient matrix in the usual sense. Then ...
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gcrd and Associates of an element of the Quaternion algebra over a totally real number field $K$
Let $K$ be a totally real number field of class number 1, and $Q$ the quaternion algebra over the ring of integers of $K$ with basis
$\{1,i,j,k\}$ such that $i^2 = j^2 = k^2 = -1$ and $ij = -ji, ik = ...
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An identity with quaternions
Let $A$ be a quaternion such that $|A|=1$.
For any quaternion $q$, define a vector
$$
\vec n(q) := \left(\rule{0pt}{5mm}\mathrm{Re}(\bar q A i \bar A),\ \mathrm{Re}(\bar qAj\bar A),\ \mathrm{Re}(\bar ...
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About $(\frac{a,1-a}{k})\cong\mathrm{M}_2(k).$
$k$ is a field.
A quaternion algebra over $k$ is a $4$-dimensional $k$-algebra with a basis $1,i,j,ij$ with the following multiplicative relations: $i^2\in k^\times, j\in k^\times, ij=-ji$ and every $...
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If multiplying real numbers with quaternions is commutative is aibj = abij valid [closed]
When you multiply out a quaternion multiplication you may see stuff like aibj, i know ij = -ji = k but what about the real numbers a and b? Is it valid to make this substitution aibj = abij?
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How to calculate the difference between quaternions [closed]
I have written some code in python. The orientation of different objects in the simulation are stored using quaternions. At one point I have some orientation q and another orientation q'. I need to ...
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Frobenius Theorem on real division algebras - proof with quaternions
I find Frobenius' Theorem on real division algebras most particularly interesting and beautiful.
I have read about some proofs, but I am looking for a proof using specifically quaternions. Does ...
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What is the derivative of unit quaternion time derivative w.r.t. to unit quaternion and angular velocity?
I am trying to get the Jacobian matrix of continuous-time rigid body dynamics using unit quaternions.
The state vector is $x=\left[p, q, v, \omega\right]$. $p, v, \omega\in\mathbb{R}^3$ are position, ...
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