PlanetMath: Hamiltonian quaternions

(more info)

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Hamiltonian quaternions

(Definition)

Definition of $\mathbb{H}$

We define a unital associative algebra $\mathbb{H}$ over $\mathbb{R}$ , of dimension 4, by the basis $\{\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}\}$ and the multiplication table

(where the element in row

and column

, not

). Thus an arbitrary element of $\mathbb{H}$ is of the form

$\displaystyle a\mathbf{1} + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}, \qquad a,b,c,d \in \mathbb{R}$

(sometimes denoted by $\left< a, b, c, d\right>$ or by $a + \left<b,c,d\right>$ ) and the product of two elements $\left<a,b,c,d\right>$ and $\left<\alpha,\beta,\gamma,\delta\right>$ (order matters) is $\left<w,x,y,z\right>$ where

$\displaystyle w$	$\displaystyle =$	$\displaystyle a\alpha - b\beta - c\gamma - d\delta$
$\displaystyle x$	$\displaystyle =$	$\displaystyle a\beta + b\alpha + c\delta - d\gamma$
$\displaystyle y$	$\displaystyle =$	$\displaystyle a\gamma - b\delta + c\alpha + d\beta$
$\displaystyle z$	$\displaystyle =$	$\displaystyle a\delta + b\gamma - c\beta + d\alpha$

The elements of $\mathbb{H}$ are known as Hamiltonian quaternions.

Clearly the subspaces of $\mathbb{H}$ generated by $\{\mathbf{1}\}$ and by $\{\mathbf{1},\mathbf{i}\}$ are subalgebras isomorphic to $\mathbb{R}$ and $\mathbb{C}$ respectively. $\mathbb{R}$ is customarily identified with the corresponding subalgebra of $\mathbb{H}$ . (We shall see in a moment that there are other and less obvious embeddings of $\mathbb{C}$ in $\mathbb{H}$ .) The real numbers commute with all the elements of $\mathbb{H}$ , and we have

$\displaystyle \lambda \cdot \left<a,b,c,d\right> = \left<\lambda a, \lambda b, \lambda c, \lambda d\right>$

for $\lambda \in \mathbb{R}$ and $\left<a,b,c,d\right> \in \mathbb{H}$ .

Norm, conjugate, and inverse of a quaternion

Like the complex numbers ( $\mathbb{C}$ ), the quaternions have a natural involution called the quaternion conjugate. If $q = a\mathbf{1} + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$ , then the quaternion conjugate of , denoted $\overline{q}$ , is simply $\overline{q} = a\mathbf{1} - b\mathbf{i} - c\mathbf{j} - d\mathbf{k}$ .

One can readily verify that if $q = a\mathbf{1} + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$ , then $q\overline{q} = (a^2 + b^2 + c^2 + d^2)\mathbf{1}$ . (See Euler four-square identity.) This product is used to form a norm $\Vert\cdot\Vert$ on the algebra (or the ring) $\mathbb{H}$ : We define $\Vert q\Vert = \sqrt{s}$ where $q\overline{q} = s\mathbf{1}$ .

If $v,w \in \mathbb{H}$ and $\lambda \in \mathbb{R}$ , then

$\Vert v\Vert \geq 0$ with equality only if $v = \left<0,0,0,0\right> = 0$
$\Vert\lambda v\Vert = \vert\lambda\vert \Vert v\Vert$
$\Vert v+w\Vert \leq \Vert v\Vert + \Vert w\Vert$
$\Vert v \cdot w\Vert = \Vert v\Vert \cdot \Vert w\Vert$

which means that $\mathbb{H}$ qualifies as a normed algebra when we give it the norm $\Vert\cdot\Vert$ .

Because the norm of any nonzero quaternion is real and nonzero, we have

$\displaystyle \frac{q\overline{q}}{\Vert q\Vert^2} = \frac{\overline{q}q}{\Vert q\Vert^2} = \left< 1, 0, 0, 0 \right>$

which shows that any nonzero quaternion has an inverse:

$\displaystyle q^{-1} = \frac{\overline{q}}{\Vert q\Vert^2}\;.$

Other embeddings of $\mathbb{C}$ into $\mathbb{H}$

One can use any non-zero to define an embedding of $\mathbb{C}$ into $\mathbb{H}$ . If $\mathbf{n}(z)$ is a natural embedding of $z \in \mathbb{C}$ into $\mathbb{H}$ , then the embedding:

$\displaystyle z \rightarrow q \mathbf{n}(z) q^{-1}$

is also an embedding into $\mathbb{H}$ . Because $\mathbb{H}$ is an associative algebra, it is obvious that:

$\displaystyle ( q \mathbf{n}(a) q^{-1} )( q \mathbf{n}(b) q^{-1} ) = q ( \mathbf{n}(a) \mathbf{n}(b) ) q^{-1}$

and with the distributive laws, it is easy to check that

$\displaystyle ( q \mathbf{n}(a) q^{-1} ) + ( q \mathbf{n}(b) q^{-1} ) = q ( \mathbf{n}(a) + \mathbf{n}(b) ) q^{-1}$

Rotations in 3-space

Let us write

$\displaystyle U=\{q\in\mathbb{H}:\vert\vert q\vert\vert=1\}$

With multiplication,

is a group. Let us briefly sketch the relation between

and the group

of rotations (about the origin) in 3-space.

An arbitrary element of can be expressed $\cos\frac{\theta}{2} + \sin\frac{\theta}{2} (a\mathbf{i} + b\mathbf{j} + c\mathbf{k})$ , for some real numbers $\theta,a,b,c$ such that . The permutation $v\mapsto qv$ of thus gives rise to a permutation of the real sphere. It turns out that that permutation is a rotation. Its axis is the line through and , and the angle through which it rotates the sphere is $\theta$ . If rotations and correspond to quaternions and respectively, then clearly the permutation $v\mapsto qrv$ corresponds to the composite rotation $F\circ G$ . Thus this mapping of onto is a group homomorphism. Its kernel is the subset $\{1,-1\}$ of , and thus it comprises a double cover of . The kernel has a geometric interpretation as well: two unit vectors in opposite directions determine the same axis of rotation.