# Is it possible a (3x3) matrix (3x1) vector multiplication represent by quaternions?

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?

• 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. – SZN Jun 22 '16 at 3:35

A quaternion $\bf q$ can be represented as a 4x4 matrix $\bf M_q$ of real values:
$${\bf q} = a+b{\bf i}+c{\bf j}+d{\bf k} : {\bf M}_{\bf q}=\left[\begin{array}{rr|rr}a&b&c&d\\-b&a&-d&c\\\hline-c&d&a&-b\\-d&-c&b&a\end{array}\right]$$
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?