adding dimensions to existing np arrays

Question

I'm trying to make a clean connection between the dimensions in a numpy array and the dimensions of a matrix via classical linear algebra. Suppose the following:

In [1]  import numpy as np
In [2]  rand = np.random.RandomState(42)
In [3]  a = rand.rand(3,2)
In [4]  a
Out[4]:
array([[0.61185289, 0.13949386],
       [0.29214465, 0.36636184],
       [0.45606998, 0.78517596]])
In [5]: a[np.newaxis,:,:]
Out[5]:
array([[[0.61185289, 0.13949386],
        [0.29214465, 0.36636184],
        [0.45606998, 0.78517596]]])
In [6]: a[:,np.newaxis,:]
Out[6]:
array([[[0.61185289, 0.13949386]],

       [[0.29214465, 0.36636184]],

       [[0.45606998, 0.78517596]]])

In [7]: a[:,:,np.newaxis]
Out[7]:
array([[[0.61185289],
        [0.13949386]],

       [[0.29214465],
        [0.36636184]],

       [[0.45606998],
        [0.78517596]]])

My questions are as follows:

Is is correct to say that the dimensions of a are 3 X 2? In other words, a 3 X 2 matrix?
Is it correct to say that the dimensions of a[np.newaxis,:,:] are 1 X 3 X 2? In other words, a matrix containing a 3 X 2 matrix?
Is it correct to say that the dimensions of a[:,np.newaxis,:] are 3 X 1 X 2? In other words a matrix containing 3 1 X 2 matrices?
Is it correct to say that the dimensions of a[:,:,np.newaxis] are 3 X 2 X1? In other words a matrix containing 3 matrices each of which contain 2 1 X 1 matrices?

As temporary bridge these descriptions may help, but in long run you should work with numpy's own terminology. Those are all arrays with varying dimensions. A numpy can have 0, 1, 2 or more dimensions. a.shape is a tuple with those dimensions, and hence can be (), (3,) or (2,3,4). Where possible numpy tries to tread each dimension as "equally-important". — hpaulj
– hpaulj, Commented Feb 12, 2021 at 20:34

Stefan B · Accepted Answer · 2021-02-12 22:34:34Z

yes
yes
yes
three 2x1 matrices each of which contains one vector of size 1

Just find out using .shape:

import numpy as np

rand = np.random.RandomState(42)

# 1.
a = rand.rand(3, 2)
print(a.shape, a, sep='\n', end='\n\n')

# 2.
b = a[np.newaxis, :, :]
print(b.shape, b, sep='\n', end='\n\n')

# 3.
c = a[:, np.newaxis, :]
print(c.shape, c, sep='\n', end='\n\n')

# 4.a
d = a[:, :, np.newaxis]
print(d.shape, d, sep='\n', end='\n\n')

# 4.b
print(d[0].shape, d[0], sep='\n', end='\n\n')
print(d[0, 0].shape, d[0, 0])

output:

(3, 2)
[[0.37454012 0.95071431]
 [0.73199394 0.59865848]
 [0.15601864 0.15599452]]

(1, 3, 2)
[[[0.37454012 0.95071431]
  [0.73199394 0.59865848]
  [0.15601864 0.15599452]]]

(3, 1, 2)
[[[0.37454012 0.95071431]]

 [[0.73199394 0.59865848]]

 [[0.15601864 0.15599452]]]

(3, 2, 1)
[[[0.37454012]
  [0.95071431]]

 [[0.73199394]
  [0.59865848]]

 [[0.15601864]
  [0.15599452]]]

(2, 1)
[[0.37454012]
 [0.95071431]]

(1,) [0.37454012]

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