Tagged Questions

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

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Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb C$,...
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Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions? Here a division algebra is an associative algebra where every ...
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Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1 & 0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
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Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
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Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
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Representing rotations using quaternions

I'm learning Unity and came across a situation where rotations are represented as Quaternions. I've heard that they where used in computer graphics, but never had to use them until now. What I can't ...
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Real world uses of Quaternions?

I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real ...
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Quaternion distance

I am using quaternions to represent orientation as a rotational offset from a global coordinate frame. Is it correct in thinking that quaternion distance gives a metric that defines the closeness of ...
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Is there an algebraic closure for the quaternions?

This post is a sequel of: Is the set of quaternions $\mathbb{H}$ algebraically closed? This answer shows that: 1. $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$...
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What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
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Rotate Quaternion A by 180 degrees

Suppose you have an arbitrary quaternion - call it A - how do you rotate it by 180 degrees? Is there a way to do this without convert to angle-axis representation, i.e., keep it within the ...
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One more question about mapping quaternionic matrices into real matrices

Real matrices that lie in the image of the inclusion homomorphism $\rho_n: M_n(\mathbb C) \to M_{2n}(\mathbb R)$ are called complex linear real matrices. It is easy to see that a real matrix is ...
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Mean value of the rotation angle is 126.5°

In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by ...
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How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
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what is the tensor product $\mathbb{H\otimes_{R}H}$

I'm looking for a simpler way of thinking about the tensor product: $\mathbb{H\otimes_{R}H}$, i.e a more known algbera which is isomorphic to it. I have built the algebra and played with it for a bit,...