Why the second dimension in a Numpy array is empty?
In case there's confusion, (3,) doesn't mean there's a missing dimension. The comma is part of the standard Python notation for a single element tuple. Shapes (1,3), (3,), and (3,1) are distinct,
While they can contain the same 3 elements, their use in calculations (broadcasting) is different, their print format is different, and their list equivalent is different:
In [21]: np.array([1,2,3])Out[21]: array([1, 2, 3])In [22]: np.array([1,2,3]).tolist()Out[22]: [1, 2, 3]In [23]: np.array([1,2,3]).reshape(1,3).tolist()Out[23]: [[1, 2, 3]]In [24]: np.array([1,2,3]).reshape(3,1).tolist()Out[24]: [[1], [2], [3]]And we don't have to stop at adding just one singleton dimension:
In [25]: np.array([1,2,3]).reshape(1,3,1).tolist()Out[25]: [[[1], [2], [3]]]In [26]: np.array([1,2,3]).reshape(1,3,1,1).tolist()Out[26]: [[[[1]], [[2]], [[3]]]]In numpy an array can have 0, 1, 2 or more dimensions. 1 dimension is just as logical as 2.
In MATLAB a matrix always has 2 dim (or more), but it doesn't have to be that way. Strictly speaking MATLAB doesn't even have scalars. An array with shape (3,) is missing a dimension only if MATLAB is taken as the standard.
numpy is built on Python which as scalars, and lists (which can nest). How many dimensions does a Python list have?
If you want to get into history, MATLAB was developed as a front end to a set of Fortran linear algebra routines. Given the problems those routines solved the concept of matrix with 2 dimensions, and row vs column vectors made sense. It wasn't until version 3.something that MATLAB was generalized to allow more than 2 dimensions (in the late 1990s).
numpy is based on several attempts to provide arrays to Python (e.g. numeric). Those developers took a more general approach to arrays, one where 2d was an artificial constraint. That has precedence in computer languages and mathematics (and physics). APL was developed in the 1960s, first as a mathematical notation, and then as a computer language. Like numpy its arrays can be 0d or higher. (Since I used APL before I used MATLAB, the numpy approach feels quite natural.)
In APL there aren't separate lists or tuples. So the shape of an array, rho A is itself an array, and rho rho A is the number of dimensions of A, also called the rank.