# 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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### Isomorphism of real quaternion algebra with the Hamiltonian quaternion.

For a given quaternion algebra $A=\left(\frac{a,b}{\mathbb{R}}\right)$ where $a,b<0$, we know that it is isomorphic to $\mathbb{H}$. Can I check if the following is the appropriate isomorphism ...
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### Effect of quaternion normalization when differentiating

I'm interested in optimizing across rotations which are represented as quaternions. You can either differentiate and then normalize after the update or you can include the normalizing term in the ...
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### Solving the quaternion sum of four squares/reduced norm

Let $K$ be an algebraic totally real number field, and let $O_K$ its ring of integers denote by $Q$ the quaternion algebra over $K$ I am looking for references on the resolution of the reduced norm ...
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### $3\times3$ real matrix decomposition to SVD using two unit quaternions and scale vector

I've been trying to search about doing $3\times3$ real matrix SVD, but instead of decomposing it into matrices, represent the two rotations as unit quaternions with the singular values as separate ...
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### Quaternions Triangular Inequality

Let $z_1, \text{ } z_2 \in \mathbb{C}$. We now that: $$\left| z_1 + z_2 \right| = \left| z_1 \right| + \left| z_2 \right| \iff \frac{z_1}{z_2} > 0 \vee \left( z_1 = 0 \vee z_2 = 0 \right)$$ My ...
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### Square Root of a Arbitrary Octonion

Let $q = a + bi + cj + dk = a + Q \in \mathbb{H}$ a quaternion. So, we have: $$\sqrt{q} = \sqrt{\frac{|q| + a}{2}} + \frac{Q}{|Q|} \sqrt{\frac{|q| - a}{2}}$$ I have a question: this formula works for ...
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### Factor a 3D rotation as a rotation along an horizontal axis followed by a rotation along the vertical

As the title says, I am trying to find, for any rotation R, a unit vector $u = [u_x, u_y, 0]$, and two angles $\alpha$ and $\beta$, such that R is equal to a rotation of angle $\alpha$ around $u$ and ...
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### Fast expression to rotate vector by the same rotation that orients one vector to another

The title is a bit convoluted but I don't know a good way to express it. I am wondering whether this expression that I've worked out for my specific task has been discussed in the past since I have ...
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### Is there a standard notation for the unit quaternions?

Of course, I mean other than $SU(2)$. I think the standard notation for the entire algebra of quaternions is $\mathbb{H}$, so I would imagine something like $\mathbb{H}^*$ or $\mathbb{H}_1$, etc. Is ...
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### What is the unitary subset of Lie algebra $su(2)$?
Lie algebra $su(2)$ consists of the $2\times 2$ skew-hermitian complex matrices with addition and multiplication by real numbers as vector space operations and commutator as Lie bracket. The i-...
I'm a math student and I'm taking an algebra course. The professor introduced us to the field of quaternions ($\mathbb{Q}$), I became very curious about the topic and I saw that in addition to ...