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Nowadays I am studying rotation using quaternion. I understand, that rotation can formulated a several way. In matrix notation: $$ \vec{v}^{new} = \bar{\bar{R}}^{new}_{old}\cdot\vec{v}^{old} $$ where $\vec{v} = \left[v_1, v_2, v_3\right]^T$ is 3d vector and $\bar{\bar{R}}$ is the rotation matrix with the size of 3x3, which transform $\vec{v}$ from the old frame to a new cordinate frame.

Or quaternion notation: $$ v^{new} = q * v^{old} * q^{-1} $$ where $v = \left(0,v_1,v_2,v_3\right)$ and $ q = \left(q_0, q_1, q_2, q_3\right)$ are quaternions. The symbol $*$ means quaternion product.

My question is that can the matrix-vector multiplication equation $$ \vec{u} = \bar{\bar{A}}\cdot\vec{v},$$ where $\vec{u}= \left[u_1, u_2, u_3\right]^T$, $\vec{v}= \left[v_1, v_2, v_3\right]^T$ 3d vectors and $\bar{\bar{A}} = a_{i,j}$ is a 3x3 matrix with 9 independent element, expressed with quaternion formalism, such that $$ u = \text{someQuaternionOperations(p,q,r,v)} $$ where $ u = \left(0, u_1, u_2, u_3\right)$, $v = \left(0, v_1, v_2, v_3\right)$, $p = \left(p_0, p_1, p_2, p_3\right)$, $q = \left(q_0, q_1, q_2, q_3\right)$, $r=\left(r_0, r_1, r_2, r_3\right) $ are quaternions?

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  • $\begingroup$ what are $p$ and $r$ supposed to be? You are working with two different representations of the rotation--the quaternion and the rotation matrix. Quaternion components can be used to calculate the elements of the rotation matrix in a well-known way--see en.wikipedia.org/wiki/Quaternions_and_spatial_rotation. Generally however, I don't think you can use quaternions to mimic any 3x3 matrix operating on a 3-vector. $\endgroup$ – SZN Jun 22 '16 at 3:35
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Sorry this got a bit too big for a comment. Not really an answer but maybe some help to get you up and running.

A quaternion $\bf q$ can be represented as a 4x4 matrix $\bf M_q$ of real values:

$${\bf q} = a+b{\bf i}+c{\bf j}+d{\bf k} : {\bf M}_{\bf q}=\left[\begin{array}{rr|rr}a&b&c&d\\-b&a&-d&c\\\hline-c&d&a&-b\\-d&-c&b&a\end{array}\right]$$

Notice that what will end up in the top row after multiplications and additions as the rows below it are just permutations sometimes with sign changes of the first row.

Note you can create a matrix ${\bf V} = [v_1,v_2,v_3]$ so that the first row in $\bf M_qV$ will be scalar products between $v_k$ and $[a,b,c,d]^T$. What would that correspond to if you had two quaternions?

Now you can start investigating for yourself where and how you can stuff functions of vector elements into a,b,c,d and how it will affect things.

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