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Say that you have an orthonormal rotation matrix

$$ R = \left[ \begin{array}{ccc} R_{xx} & R_{xy} & R_{xz} \\ R_{yx} & R_{yy} & R_{yz} \\ R_{zx} & R_{zy} & R_{zz} \end{array} \right] $$

This matrix rotates a $3 \times N$ matrix of 3-dimensional column vectors $A$ in some arbitrary way about the X, Y, and Z axis as $A' = RA$. Starting from this point, how can I remove a rotation about an axis?

For example, this arises if I want to preserve a certain direction. Let's say my column vectors in the $A$ matrix amount to points in 3D space. They are all oriented on some plane with a normal vector $n$. I want to modify $R$ in such a way that the my modified rotation matrix $R'$ applies the transformation to my set of points $A$ without affecting my normal $n$. Put more simply, how can I modify $R$ to ensure that $A'$ and $A$ have the same normal vector $n$?

For the sake of argument, we can assume that $n = [0, 1, 0]$.

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    Instead of modifying $R$, couldn't you just choose it more judiciously so that its axis of rotation is $n$ (equivalently its plane of rotation is $n^\perp$)? – 2017-02-26
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    @Bye_World: In my case no. I am estimating R using a least squares method as a transformation between two coordinate systems. However, for my use case, I can throw out one of the directions. In essence I guess I am transforming my initial points $A$ to $A'$, representing a different coordinate system. Then subsequently, I'm rotating the new coordinate system so that $n$ (the plane normal from $A$) matches to $n'$ (the plane normal in the new coordinate system) – 2017-02-26
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    Rotations about axes are not commutative, so you will need to decide which axis **and** which order of rotation you want to split out. – 2017-02-26

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