# What is the equivalent of a (non-identity) matrix in quaternions

I'm using quaternions to solve Euler's equations of motion. I have the substitution that $$\dot{q} = \frac{1}{2}q\bigodot\omega$$, where $$\omega$$ is the angular velocity of the rotating frame. Hence $$\omega = 2q^*\bigodot \dot{q}$$ where $$q^*$$ is the conjugate of $$q$$. In Euler's quation, the inertia tensor normally multiplies the $$\omega$$ vector like so: $$\omega \times I\omega$$. I understand that in the setting of quaternions the vector $$\omega$$ becomes $$\omega_{[quat]} = [0, \omega_x\bf{i}, \omega_y\bf{j}, \omega_z\bf{k}]$$ and is said to have no real part. But how do we express the inertia tensor for this new convention to work? I was considering treating it as a rotation matrix and converting to a quaternion itself but this doesn't work when det$$(I)\neq1$$ where $$I$$ is the inertia tensor.

Any help greatly appreciated.

Render the following as quaternion imaginary units:

$$\sigma_x=\begin{bmatrix}i&0\\0&-i\end{bmatrix}$$

$$\sigma_y=\begin{bmatrix}0&i\\i&0\end{bmatrix}$$

$$\sigma_z=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$

And the real unit matrix:

$$\sigma_0=\begin{bmatrix}1&0\\0&1\end{bmatrix}$$

with $$i$$ as the imaginary unit in ordinary complex numbers. Then for all real numbers $$a,b,c,d$$ the combination

$$a\sigma_0+b\sigma_x+c\sigma_y+d\sigma_z$$

will reproduce all the algebraic properties of quaternions with "real part" $$a$$ and "imaginary parts" $$b,c,d$$. For instance, $$(\sigma_x)(\sigma_y)=-(\sigma_y)(\sigma_x)=\sigma_z$$ and $$(\sigma_x)^2=(\sigma_y)^2=(\sigma_z)^2=-\sigma_0$$.

• Thanks for this response, although I'm not entirely sure how I would use this to transform $I$ into a quaternion. Are you suggesting I essentially express the quaternions as matrices instead? Thanks Nov 2 '18 at 1:40
• That was how I interpreted it based on the title. Maybe something was unclear? Nov 2 '18 at 1:41
• Sorry, to rephrase it: I have an inertia tensor that is 3x3 and normally multiplies a 3x1 vector. However I've transformed my 3x1 vector into a quaternion by adding 0 real part. What must I do to the inertia tensor for this to still be a calculable multiplication. Nov 2 '18 at 2:26