How to switch $y$ and $z$ axis of a rotation matrix very simple question how to switch the $y$ and $z$-axis of a rotation matrix.
So far I have rotated the rotation matrix $90$ degrees around $x$. The only thing left would be to inverse the new $y$ axis, which is the old $z$ axis. How can I do this? Or is there a way to do the switch in one step?
Thanks!
 A: I'm not sure what "one step" means but simple conjugation by the permutation matrix
$$
P_{23}=\left(
\begin{array}{ccc}
 1 & 0 & 0 \\
 0 & 0 & 1 \\
 0 & 1 & 0 \\
\end{array}
\right)
$$
will do the trick.  This permutation amounts to interchanging the $y$ and $z$ axis.  The matrix itself if not in $SO(3)$ (its determinant is -1) but by conjugating your rotation you introduce two such factors with the result that, if
$$
R_y(\theta)=\left(
\begin{array}{ccc}
 \cos (\theta ) & 0 & -\sin (\theta ) \\
 0 & 1 & 0 \\
 \sin (\theta ) & 0 & \cos (\theta ) \\
\end{array}
\right)
$$
then 
$$
P^{-1}_{23} R_y(\theta)P_{23}=R_{P_{23}y}(\theta)=R_z(-\theta)\in SO(3)\, .
$$
Note that 
$$
R_x(\pi/2)=\left(
\begin{array}{ccc}
 1 & 0 & 0 \\
 0 & 0 & 1 \\
 0 & -1 & 0 \\
\end{array}
\right)
$$
is a proper rotation so that $R^{-1}_x(\pi/2)R_y(\theta)R_x(\pi/2)$ will produce the matrix $R_z(\theta)$. 
A: Not really sure what you are trying to do, so I will just state the rotation matricies for you.
Rotations about the $x$- axis
\begin{eqnarray*}
R_x(r)= \left(
\begin{array}{ccc} 
1 & 0 & 0  \\
0 & \cos(r) & \sin(r) \\
0 & -\sin(r) & \cos(r) \\ 
\end{array} \right)
\end{eqnarray*}
Rotations about the $y$-axis
\begin{eqnarray*}
R_y(p)= \left(
\begin{array}{ccc} 
\cos(p) & 0 & -\sin(p)  \\
0 & 1 & 0 \\
\sin(p) & 0 & \cos(p) \\ 
\end{array} \right)
\end{eqnarray*}
Rotations about the $z$-axis
\begin{eqnarray*}
R_z(y) = \left(
\begin{array}{ccc}
\cos(y) & \sin(y) & 0  \\
-\sin(y) & \cos(y) & 0 \\ 
0 & 0 & 1 \\
\end{array} \right)
\end{eqnarray*}
