Why is the product of two rotation matrices not commutative?

Question

Is there any intuition why rotational matrices are not commutative? I assume the final rotation is the combination of all rotations. Then how does it matter in which order the rotations are applied?

Answer 1

Here is a picture of a die:

Now let's spin it $90^\circ$ clockwise. The die now shows

After that, if we flip the left face up, the die lands at

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Now, let's do it the other way around: We start with the die in the same position:

Flip the left face up:

and then $90^\circ$ clockwise

If we do it one way, we end up with $3$ on the top and $5, 6$ facing us, while if we do it the other way we end up with $2$ on the top and $1, 3$ facing us. This demonstrates that the two rotations do not commute.

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Since so many in the comments have come to the conclusion that this is not a complete answer, here are a few more thoughts:

• Note what happens to the top number of the die: In the first case we change what number is on the left face, then flip the new left face to the top. In the second case we first flip the old left face to the top, and then change what is on the left face. This makes two different numbers face up.
• As leftaroundabout said in a comment to the question itself, rotations not commuting is not really anything noteworthy. The fact that they do commute in two dimensions is notable, but asking why they do not commute in general is not very fruitful apart from a concrete demonstration.

Answer 2

Matrices commute if they preserve each others' eigenspaces: there is a set of eigenvectors that, taken together, describe all the eigenspaces of both matrices, in possibly varying partitions.

This makes intuitive sense: this constraint means that a vector in one matrix's eigenspace won't leave that eigenspace when the other is applied, and so the original matrix's transformation still works fine on it.

In two dimensions, no matter what, the eigenvectors of a rotation matrix are $[i,1]$ and $[-i,1]$. So since all such matrices have the same eigenvectors, they will commute.

But in three dimensions, there's always one real eigenvalue for a real matrix such as a rotation matrix, so that eigenvalue has a real eigenvector associated with it: the axis of rotation. But this eigenvector doesn't share values with the rest of the eigenvectors for the rotation matrix (because the other two are necessarily complex)! So the axis is an eigenspace of dimension 1, so rotations with different axes can't possibly share eigenvectors, so they cannot commute.

Answer 3

In 3D space why they are not commutative?

Because you can exhibit two rotations $a,b$ such that $a\circ b\neq b\circ a$.

Take, for example, $a$ to be a rotation of $90$ degrees counterclockwise around the $x$ axis and $b$ to be a rotation of $90$ degrees counterclockwise around the $y$ axis.

Doing $a\circ b$ maps the $x$ axis onto the $y$ axis, but $b\circ a$ maps the $x$-axis onto the $z$ axis.

Answer 4

Here is a pictorial explanation equivalent to Arthur's answer:

(Picture source: Benjamin Crowell, General Relativity, p. 256.)

Answer 5

Imagine yourself walking a narrow bridge across a deep canyon. You stop and rotate face down onto the bridge, then rotate on your side to watch the beautiful sunset at the far end of the valley. By that time, however, someone who would have done the very same rotations, only in the opposite order, would be lying face down at the bottom of the canyon.

Answer 6

Consider the group of permutations of 3 objects in a line as generated by two 'rotations' $r,s$, where $r$ swaps the first two in the line and $s$ swaps the last two in the line.

$r(1,2,3) = (2,1,3)$.

$s(1,2,3) = (1,3,2)$.

Note that $rs \ne sr$ since:

$rs(1,2,3) = (3,1,2)$.

$sr(1,2,3) = (2,3,1)$.

Now you may ask, what has this got to do with 'real rotations' in space? In fact this is precisely the same phenomenon as the dice example given by Arthur. The 3 objects are the 3 orthogonal (undirected) axes that are perpendicular to the faces, and the two rotations mentioned indeed swap different pairs of axes!

Answer 7

Picture the $x$-, $y$-, and $z$-axes in the positions in which they are conventionally drawn.

Rotate a cube $90^\circ$ counterclockwise with the $x$-axis as the axis of rotation.

Then rotate it $90^\circ$ counterclockwise with the $y$-axis as the axis of rotation.

After those two rotations, the $x$-axis now points in the negative-$z$ direction, the $z$-axis in the negative-$y$ direction, and the $y$-axis in the positive-$x$ direction.

Now do the same two rotations in the opposite order. You'll get a different result.

Here it is essential that the two axes about which we rotated were different. Rotations in $3$-dimensional space that are both about the same axis commute with each other.

Answer 8

I'm discussing here non-commutativity of rotations made about current axes..

Rotations in 3d are non commutative because rotation changes direction of every potential other axis except itself (unlike in 2d where it is nothing to change because it is only one "axis" of rotation - it can be reduced in 3D to rotation about Z axis ).

Denote succession of rotations about Z-axis as $Z_1, Z_2,... Z_n$ (it's convenient to imagine that there is multitude of them by a small angular increment let's say $1^\circ$ or $2^\circ$ ) and one rotation about different axis $X$ ( it can be also not very big). When we have only rotations $..Z_j...$ we can change order of rotations in any way - the result will be the same.

But if we insert rotation $\mathbf{X}$ somewhere into middle of rotations succession $Z_1,{Z_2}...{Z_n}$ the final result $Z_1,{z_2}...{Z_j}{\mathbf{X}} {Z_k}...{Z_n}$ now depends on which place we inserted it because inserting it means that we are changing current axis of rotations for all successive rotations Z - the current Z axis, after this operation, is changed in the global frame.

It could be seen the other effect of inserting X rotation into different places of succession of $ Z_i $ if we would track the end of X versor (vector $i$) :

• If this X rotation is made at the beginning the plane XY constant in current frame for Z rotations is changed and the endpoint of X axis is distancing itself from original XY plane with every successive incremental rotation about changed now Z-axis.

• If it would be made at the end, the distance from original XY plane still would be equal $0$ !

  As the versors $i,j,k$ after whole linear operation represents the resulting matrix we clearly see that these two operations are quite different. Graphically one can imagine this track of the endpoint of $X$ versor on the surface of the unit sphere.

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The simplest alternative presentation of non-commutativity in 3D (approach here is more algebraic than geometric as previous one) can be made using axis unit vector $v_1$ of rotation, say, $R_1$.

($R_2$, we assume, has different axis with vector $v_2 \ne v_1$ and moreover we assume $R_2(v_1) \ne -v_1$ which together with $R_2(v_1) \ne v_1$ states that transformed by $R_2$ vector $v_1$ is not lying in the axis of $R_1$).

Apply it to $R_2R_1(v_1)=R_2(v_1) = v_{21}$. (Obviously $v_{21} \ne v_1$).

In reverse order $R_1R_2(v_1)=R_1(v_{21} )$.
But $v_{21} $ is not lying in the axis of $R_1$ !
Hence the result must be different than $v_{21}$ and $R_1R_2 \ne R_2R_1$.

They commute only if they share common axis or in the case of different axes they preserve each other axes with result vector changing sign i.e. $R_2(v_1)=-v_1$ and $R_1(v_2)=-v_2$ what is possible when $v_1$ is orthogonal to $v_2$ and rotations are by $\pi$ angle.

Answer 9

Many comments on the original post pondered if this is a good question because "why should we expect them to be commutative in the first place?"

I think it's a great question for the following reason. One major purpose of science is to make what seems mysterious intuitive. Diseases aren't actually caused by things we can't hope to understand like the whim of the gods or "bad humors in the blood." Heavier objects fall more quickly because of air resistance, not gravity. The goal is prediction, but one result is increased intuition.

Let's take the last example. Someone who had never heard of Galileo and the Tower of Pisa, had never heard of physics, etc. could very easily ask the question "is there any reason why in a vacuum all objects should fall at the same rate [i.e. accelerate the same]?" One could replay "well, what reason would you have for it being any other way?" Philosophically true, but not helpful.

The reason this example isn't intuitive is that we never experience this "same rate" phenomenon in everyday life. Heavier things pull us down more, are better at holding things in place, etc. So it's more intuitive that they should fall faster. You see where I'm going with this.

Another great example from physics would be momentum. When have you ever pushed something and it goes forever?

So looking at some of the examples that others have given demonstrating that rotations don't commute, it's interesting to think about why the fact does not become intuitive given how often we manipulate objects in 3d space?

So, let's address the question:

Is there any intuition why rotational matrices are not commutative? I assume the final rotation is the combination of all rotations. Then how does it matter in which order the rotations are applied?

1. While we manipulate objects in 3D all day long, we don't do that nearly as much as in 2D. We mostly work on surfaces (counters, workbenches, floors, walls, etc.). In that context, rotations do commute---order does not matter.

2. Most objects we actually use in 3D are so asymmetrical and directed that we never have to consider compound rotations. A fork has only one way to hold it that makes sense. If you want to use it, you grab the handle and point the tines downward. You don't sit there doing compound rotations on it. When was the last time you manipulated a die or a book or a bed by doing compound rotations? I usually just lay a book down with the cover facing up, the binding on the left, and open it. I don't often rotate my bed and fall forward onto it :) So what experiences would have given me a chance to build intuition

3. There are some situations where we might actually consider compound rotations, such as moving furniture and loading it into a truck with our partner. If you have a romantic partner and have ever moved furniture and loaded it into a truck, you will agree that this is not exactly an intuition-building experience.

4. For someone who assembles machinery or puts together pipe fittings, it may be extremely intuitive that rotations do not commute. If you asked a person in such a field "if I rotate this 3-say plumbing fitting vertically and then horizontally vs the opposite way, will it end up in the same orientation," and they weren't too busy working to answer your question, they would likely know the correct answer.

To summarize, the non-commutativity of compound rotations is not intuitive because we just don't have a lot of life experiences where it is important or useful. Meanwhile, non-commutative operations are VERY familiar to us (just think about it). So if you go through a number of the exercises others have shown above, you will soon find that it is indeed very intuitive.