Re: [Algorithms] Rotation Matrices

[Ron L. <ro...@do...>]

Nik...@ao... wrote

> Why isn't the multiplication of rotational matrices commutative?

Because concatenation of rotations is not commutative.  In general,
concatenation of mappings is not commutative.

>For example
> if I have one rotation matrix which rotates 20 degrees around the Z axis,
and
> another which rotates 50 degrees around the X axis, depending on the order
in
> which I multiply them, I get different results.
>Why is this?  When rotating
> an object there is no visible difference when it is first rotated on one
axis
> and then on another, is there?

Wrong. Try a simple experiment to see for yourself.

Hold in your hand a rectangular box, whose three edges have different
lengths or are otherwise easily distinguishable.  Now rotate the box 90
degrees about the x -axis, then 90 degress about the y-axis.  Notice how it
is oriented.  Now return it to its original orientation and rotate first 90
degrees about the y-axis, then 90 degrees about the x-axis.  Notice that you
end up in a different final orientation.
That's the way the world is, in 3D anyway.  Rotations are not commutative.
If matrix multiplication were commutative then matrices would not be
suitable for representing rotations.

Concatenation of mappings in general is  not commutative, even 1-dimensional
mappings.  For example consider the 1-1 mapping of the line to itself given
by
x->x^3.  Now consider another 1-1 mapping of the line to itself given by
x->x+1.  Now concatenate them in two different orders.  In one case you get
x->(x+1)^3 in the other case you get
x->x^3+1.
As functions x^3 + 1 and (x +1)^3 = x^3 + 3x^2 + 3x + 1  are quite
different.  Concatenation of mappings, in general, is not commutative.

>I first looked in an elementary linear
> algebra book for the answer; however it stated that the multiplication of
> rotation matrices is commutative (although it was only covering 2D
rotation
> matrices). Any help would be appreciated.
>

It is true that concatenation of rotations about the origin in the plane is
indeed commutative.   That's just the way the world is.