Let's assume that we want to find a rotation matrix which added to a given rotation matrix gives also a rotation matrix. I would name such matrix a rotation additive matrix for a given rotation matrix.
First consider a 2D case for identity matrix. It is relatively easy to find such matrix.
$
R= \begin{bmatrix}
-\dfrac{1}{2} & -\dfrac {\sqrt{3}}{2} \\
\dfrac{\sqrt{3}}{2} & -\dfrac{1}{2} \\
\end{bmatrix}
$
Really we have
$
\begin{bmatrix}
-\dfrac{1}{2} & -\dfrac { \sqrt{3}}{2} \\
\dfrac{ \sqrt{3}}{2} & -\dfrac{1}{2} \\
\end{bmatrix} + \begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix} = \begin{bmatrix}
\dfrac{1}{2} & -\dfrac { \sqrt{3}}{2} \\
\dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\
\end{bmatrix}
$
Also symmetrical matrix to R is additive for identity matrix, so we have at least 2 such matrices. If it exists for identity matrix should, I believe, exist for other 2D rotation matrices.
I was searching also for a such matrices in 3D. However without positive effects.
Question
Do such matrices exist in 3D ?
- If so how to find them.
- If not how to prove it.