clear authoritative explanation of numpy axis numbers?

Question

I am getting confused by contradictory explanations of what exactly the term axis means in numpy and how these constructs are numbered.

Here's one explanation:
Axes are defined for arrays with more than one dimension.
A 2-dimensional array has two corresponding axes:
the first running vertically downwards across rows (axis 0), and
the second running horizontally across columns (axis 1).

So, in this 3x4 matrix ...

>>> b = np.arange(12).reshape(3,4)
>>> b
array([[ 0,  1,  2,  3],
       [ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])

(axis 0) is the 3 rows
(axis 1) is the 4 columns

So the rule might be ...

In an MxN matrix, (axis 0) is M and (axis 1) is N.

Is this correct?

So, in a 3 dimensional matrix AxBxC (axis 0) is A
(axis 1) is B
(axis 2) is C

Is this correct?

Answer 1

Everything you said is correct, with the exception of

Axes are defined for arrays with more than one dimension.

Axes are also defined for one dimensional arrays - there is just one of them (i.e. axis 0).

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One intuitive way to think about axes is to consider what happens when you apply a reduction operation over one axis, such as summation. For example, suppose I have some array x:

x = np.arange(60).reshape(3, 4, 5)

If I compute x.sum(0) I am "collapsing" x over the first dimension (i.e. axis 0), so I end up with a (4, 5) array. Likewise, x.sum(1) gives me a (3, 5) array and x.sum(2) gives me a (3, 4) array.

An integer index into a single axis of x will also give me an output with one fewer axis. For example, x[0, :, :] gives me the first "row" of x, which has shape (4, 5), x[:, 0, :] gives me the first "column" with shape (3, 5), and x[:, :, 0] gives me the first slice in the third dimension of x with shape (3, 4).