# What Happens When I Change the Order of Multiplication of a Rotation Matrix

Given an arbitrary 3x3 matrix M and an angle θ I can use this to, we'll say rotate about my space's Z-axis, so I'd create the rotation matrix:

$$R_z = \begin{bmatrix} \cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1 \end{bmatrix}$$

What I'm struggling with is the description of these 2 multiplications:

1. M*Rz
2. Rz*M

I believe that 1 describes M rotated about my space's Z-axis and 2 describes rotating M about it's Z-axis. Is this correct? Can someone help me with understanding this?

• I assume * represents matrix multiplication (instead of complex conjugation for example)? – YiFan Sep 11 at 13:42
• @YiFan Yes, would it be more clear if I just wrote MR? I can change it if you think so? – Jonathan Mee Sep 11 at 13:46

In general, for $$n\times n$$ matrices $$A,B$$, the matrix product $$AB$$ represents the composite map obtained by applying $$B$$ first then applying $$A$$. This is simply because for any vector $$v$$, $$ABv=A(Bv)$$, so multiplication by $$B$$ is done first.
I believe that $$1$$ describes $$M$$ rotated about my space's $$Z$$-axis and $$2$$ describes rotating $$M$$ about it's $$Z$$-axis.
No, this is not correct the way it is written. What does it even mean for a matrix to be "rotated" about an axis? It is however true that $$MR_z$$ corresponds to the linear transformation which first rotates the space about the $$z$$-axis, and then applies the linear transformation defiend by $$M$$. On the other hand, $$R_zM$$ corresponds to applying $$M$$ first, and then rotating about the $$z$$-axis. In general, these two are not the same, of course.
It would also be correct to say that $$R_zM$$ has column vectors which are rotated copies of the column vectors of $$M$$. That is, if $$M=\begin{bmatrix}\vert&\vert&\vert\\v_1&v_2&v_3\\\vert&\vert&\vert\end{bmatrix},$$ then $$R_zM=\begin{bmatrix}\vert&\vert&\vert\\R_zv_1&R_zv_2&R_zv_3\\\vert&\vert&\vert\end{bmatrix}.$$ Of course, you will observe that $$R_zv_i$$ is just $$R_z$$ applied to $$v_i$$, which corresponds to $$v_i$$ rotated about the $$z$$-axis (for each $$i=1,2,3$$). Furthermore, $$MR_z$$ has row vectors which are rotated copies of the row vectors of $$M$$. That is, if $$M=\begin{bmatrix}-&u_1&-\\-&u_2&-\\-&u_3&-\end{bmatrix}$$ then $$MR_z=\begin{bmatrix}-&u_1R_z&-\\-&u_2R_z&-\\-&u_3R_z&-\end{bmatrix}.$$ Of course, you will again recognise that $$u_iR_z$$ is just a rotated copy of the $$u_i$$, for each $$i=1,2,3$$.