> Although length-1 axes already being in the data sounds worrying: you have an array that doesn't depend on an axis, but there's an axis anyway to show where it would go if it did?

Example:

    A / A.sum(axis=3, keepdims=True)

Make all of the vectors along axis=3 sum up to one. There are other ways of doing it, but this way seems fairly clear to me. Unfortunately for this example (and several others!) keepdims=True isn't the default. keepdims=True is the behavior in Matlab/Octave, so the example becomes

    A / sum(A, 4)

with 4=3+1 because Matlab is 1-based like Fortran.