Making a random rotation matrix is somewhat hard. You can’t just use “random elements”; that’s not a random matrix.
First attempt: Rotate around a random vector
My first thought was the following:
Pick a random axis \(\hat u\), by getting three Gaussian-distributed numbers, calling them x, y, and z, and then taking the norm of that vector.
Pick a random angle in the range \(0 \le \theta < 2 \pi\).
Rotate your original vector \(\vec v\) around \(\hat u\) by \(\theta\).
That’s accomplished with the following code:
Unfortunately, this does not give you a uniformly rotated vector: you end up where you started too often.
Second attempt: Uniform Random Rotation Matrix
If we want a uniformly distributed rotation, that’s a little trickier than the above. There’s a Wikipedia article section on it, but its not too helpful.
Instead, I found some C code from a book called “Graphics Gems III”.
Then, I started with a direct translation of the C code:
Then I did a more Pythonic version, using numpy arrays more to their potential, and adding an option to use pre-generated random numbers:
This method does a much better job of creating a uniform distribution. In other words, for any vector \(\vec v\), after multiplying \(\vec v^\prime = M \vec v\) (or v2 = M.dot(v)), the new vector \(\vec v^\prime\) will be uniformly distributed over the sphere.