Difference between revisions of "Basic properties of quaternions"

Revision as of 16:23, 4 September 2015

The imaginary units of a quaternion are defined as follows:

i^2 = j^2 = k^2 = ijk = -1

Unlike the multiplication of real or complex numbers, the multiplication is not commutative:

\begin{align} ij &= -ji = k \\ jk &= -kj = i \\ ki &= -ik = j \end{align}

The conjugate $\mathbf{q}^*$ of a quaternion $\mathbf{q}$ is obtained by negating the imaginary or vector part, respectively:

q^*= \text{Re}(q) - \text{Im}(q) = q_0-\mathbf{q} = q_0 - q_1i - q_2j - q_3k

Multiplying a quaternion with its conjugate like described in the subarticle about multiplication of quaternions consequently leads to

qq^* = q^*q =q_0^2 + q_1^2 + q_2^2 + q_3^2

The conjugate can be used to easily determine the real part and the imaginary part of a quaternion:

\begin{align} \text{Re}(q) &= \frac{1}{2}(q+q^*) \\ \text{Im}(q) &= \frac{1}{2}(q-q^*) \end{align}

The most important properties of the conjugate are:

\begin{align} (q^*)^* &= q \\ (p+q)^* &= p^* + q^* \\ (pq)^* &= q^*p^* \\ q^* &= -\frac{1}{2}(q+iqi+jqj+kqk) \end{align}

The norm of a quaternion corresponds to its euclidean length in four-dimensional space and is computed as follows:

|q| = \sqrt{qq^*} = \sqrt{q_0^2 + q_1^2 + q_2^2 + q_3^2}

It holds:

|pq| = |p||q|

The inverse of a quaternion is defined as follows:

q^{-1} = \frac{1}{q} = \frac{q^*}{|q|^2}

This can be proven by multiplying a quaternion $q$ with its inverse using the definition of the norm:

qq^{-1} = q\frac{q^*}{|q|^2} = \frac{qq^*}{qq^*} = 1

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 qq^{-1} = q\frac{q^*}{|q|^2} = \frac{qq^*}{qq^*} = 1
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