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

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

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

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

Define

$$ H = a[1]+b[i]+c[j]+d[k]$$

$$ H = \begin{bmatrix} a+di & -b-ci \\ b-ci & a-di \\ \end{bmatrix} $$

$$ H = \begin{bmatrix} z & w \\ -\bar w & \bar z \\ \end{bmatrix} $$

where $$z= a+di$$ $$w=-b-ci$$

This is the basic quaternion setup we've been taught, my question is the following:

Define $\alpha$, $\beta$ are matrices in the form of $H$. What is $\alpha \cdot \beta$?

For example before we even get to multiplying the two H matrices, what does one H matrix represent in terms of a geometric interpretation? Is it a rotation in $R^4$? Am I even interpreting what the question is asking properly?

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    $\begingroup$ What's wrong with using regular matrix multiplication to find $\alpha\cdot\beta$? They have chosen to represent these as matrices for a reason. $\endgroup$ – Arthur Aug 17 at 7:36
  • $\begingroup$ I think I worded it wrong, $\alpha \beta$ are matrices in the form of $H$, what is the result of multiplying them together. It's just a question the prof has put down and I really can't piece together what he's trying to ask here. $\endgroup$ – Krio Aug 17 at 7:41
  • $\begingroup$ Looks like he's asking for $\begin{bmatrix}z_\alpha & w_\alpha \\ -\bar w_\alpha & \bar z_\alpha \end{bmatrix}\cdot \begin{bmatrix}z_\beta & w_\beta \\ -\bar w_\beta & \bar z_\beta \end{bmatrix}$. And $H$ is just a quaternion -- no special geometric interpretation other than a point in 4-dimensional space. $\endgroup$ – Klaas van Aarsen Aug 17 at 9:30
  • $\begingroup$ thank you for this $\endgroup$ – Krio Aug 17 at 10:16

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