Explaining the differences between dim, shape, rank, dimension and axis in numpy

Question

I'm new to python and numpy in general. I read several tutorials and still so confused between the differences in dim, ranks, shape, aixes and dimensions. My mind seems to be stuck at the matrix representation. So if you say that A is a matrix that looks like this:

A = 

1 2 3
4 5 6

then all I can think of is a 2x3 matrix (two rows and three columns). Here I understand that the shape is 2x3. But I really I am unable to go out side the thinking of a 2D matrices. I don't understand for example the dot() documentation when it says "For N dimensions it is a sum product over the last axis of a and the second-to-last of b". I'm so confused and unable to understand this. I don't understand like if V is a N:1 vector and M is N:N matrix, how dot(V,M) or dot(M,V) work and the difference between them.

Can anyone then please explain to me what is a N dimensional array, what's a shape, what's an axis and how does it relate to the documentation of the dot() function? It would be great if the explanation visualizes the ideas.

Answer 1

Dimensionality of NumPy arrays must be understood in the data structures sense, not the mathematical sense, i.e. it's the number of scalar indices you need to obtain a scalar value.(*)

E.g., this is a 3-d array:

>>> X = np.arange(24).reshape(2, 3, 4)
>>> X
array([[[ 0,  1,  2,  3],
        [ 4,  5,  6,  7],
        [ 8,  9, 10, 11]],

       [[12, 13, 14, 15],
        [16, 17, 18, 19],
        [20, 21, 22, 23]]])

Indexing once gives a 2-d array (matrix):

>>> X[0]
array([[ 0,  1,  2,  3],
       [ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])

Indexing twice gives a 1-d array (vector), and indexing three times gives a scalar.

The rank of X is its number of dimensions:

>>> X.ndim
3
>>> np.rank(X)
3

Axis is roughly synonymous with dimension; it's used in broadcasting operations:

>>> X.sum(axis=0)
array([[12, 14, 16, 18],
       [20, 22, 24, 26],
       [28, 30, 32, 34]])
>>> X.sum(axis=1)
array([[12, 15, 18, 21],
       [48, 51, 54, 57]])
>>> X.sum(axis=2)
array([[ 6, 22, 38],
       [54, 70, 86]])

To be honest, I find this definition of "rank" confusing since it matches neither the name of the attribute ndim nor the linear algebra definition of rank.

Now regarding np.dot, what you have to understand is that there are three ways to represent a vector in NumPy: 1-d array, a column vector of shape (n, 1) or a row vector of shape (1, n). (Actually, there are more ways, e.g. as a (1, n, 1)-shaped array, but these are quite rare.) np.dot performs vector multiplication when both arguments are 1-d, matrix-vector multiplication when one argument is 1-d and the other is 2-d, and otherwise it performs a (generalized) matrix multiplication:

>>> A = np.random.randn(2, 3)
>>> v1d = np.random.randn(2)
>>> np.dot(v1d, A)
array([-0.29269547, -0.52215117,  0.478753  ])
>>> vrow = np.atleast_2d(v1d)
>>> np.dot(vrow, A)
array([[-0.29269547, -0.52215117,  0.478753  ]])
>>> vcol = vrow.T
>>> np.dot(vcol, A)
Traceback (most recent call last):
  File "<ipython-input-36-98949c6de990>", line 1, in <module>
    np.dot(vcol, A)
ValueError: matrices are not aligned

The rule "sum product over the last axis of a and the second-to-last of b" matches and generalizes the common definition of matrix multiplication.

(*) Arrays of dtype=object are a bit of an exception, since they treat any Python object as a scalar.

Answer 2

np.dot is a generalization of matrix multiplication. In regular matrix multiplication, an (N,M)-shape matrix multiplied with a (M,P)-shaped matrix results in a (N,P)-shaped matrix. The resultant shape can be thought of as being formed by squashing the two shapes together ((N,M,M,P)) and then removing the middle numbers, M (to produce (N,P)). This is the property that np.dot preserves while generalizing to arrays of higher dimension.

When the docs say,

"For N dimensions it is a sum product over the last axis of a and the second-to-last of b".

it is speaking to this point. An array of shape (u,v,M) dotted with an array of shape (w,x,y,M,z) would result in an array of shape (u,v,w,x,y,z).

---

Let's see how this rule looks when applied to

In [25]: V = np.arange(2); V
Out[25]: array([0, 1])

In [26]: M = np.arange(4).reshape(2,2); M
Out[26]: 
array([[0, 1],
       [2, 3]])

First, the easy part:

In [27]: np.dot(M, V)
Out[27]: array([1, 3])

There is no surprise here; this is just matrix-vector multiplication.

Now consider

In [28]: np.dot(V, M)
Out[28]: array([2, 3])

Look at the shape of V and M:

In [29]: V.shape
Out[29]: (2,)

In [30]: M.shape
Out[30]: (2, 2)

So np.dot(V,M) is like matrix multiplication of a (2,)-shaped matrix with a (2,2)-shaped matrix, which should result in a (2,)-shaped matrix.

The last (and only) axis of V and the second-to-last axis of M (aka the first axis of M) are multiplied and summed over, leaving only the last axis of M.

If you want to visualize this: np.dot(V, M) looks as though V has 1 row and 2 columns:

[[0, 1]] * [[0, 1],
            [2, 3]] 

and so, when V is multiplied by M, np.dot(V, M) equals

[[0*0 + 1*2],     [2, 
 [0*1 + 1*3]]   =  3] 

However, I don't really recommend trying to visualize NumPy arrays this way -- at least I never do. I focus almost exclusively on the shape.

(2,) * (2,2)
   \   /
    \ /
    (2,)

You just think about the "middle" axes being dotted, and disappearing from the resultant shape.

---

np.sum(arr, axis=0) tells NumPy to sum the elements in arr eliminating the 0th axis. If arr is 2-dimensional, the 0th axis are the rows. So for example, if arr looks like this:

In [1]: arr = np.arange(6).reshape(2,3); arr
Out[1]: 
array([[0, 1, 2],
       [3, 4, 5]])

then np.sum(arr, axis=0) will sum along the columns, thus eliminating the 0th axis (i.e. the rows).

In [2]: np.sum(arr, axis=0)
Out[2]: array([3, 5, 7])

The 3 is the result of 0+3, the 5 equals 1+4, the 7 equals 2+5.

Notice arr had shape (2,3), and after summing, the 0th axis is removed so the result is of shape (3,). The 0th axis had length 2, and each sum is composed of adding those 2 elements. The shape (2,3) "becomes" (3,). You can know the resultant shape in advance! This can help guide your thinking.

To test your understanding, consider np.sum(arr, axis=1). Now the 1-axis is removed. So the resultant shape will be (2,), and element in the result will be the sum of 3 values.

In [3]: np.sum(arr, axis=1)
Out[3]: array([ 3, 12])

The 3 equals 0+1+2, and the 12 equals 3+4+5.

---

So we see that summing an axis eliminates that axis from the result. This has bearing on np.dot, since the calculation performed by np.dot is a sum of products. Since np.dot performs a summing operation along certain axes, that axis is removed from the result. That is why applying np.dot to arrays of shape (2,) and (2,2) results in an array of shape (2,). The first 2 in both arrays is summed over, eliminating both, leaving only the second 2 in the second array.

Answer 3

In your case,

1. A is a 2D array, namely a matrix, with its shape being (2, 3). From docstring of numpy.matrix:

   A matrix is a specialized 2-D array that retains its 2-D nature through operations.

2. numpy.rank return the number of dimensions of an array, which is quite different from the concept of rank in linear algebra, e.g. A is an array of dimension/rank 2.

3. np.dot(V, M), or V.dot(M) multiplies matrix V with M. Note that numpy.dot do the multiplication as far as possible. If V is N:1 and M is N:N, V.dot(M) would raise an ValueError.

e.g.:

In [125]: a
Out[125]: 
array([[1],
       [2]])

In [126]: b
Out[126]: 
array([[2, 3],
       [1, 2]])

In [127]: a.dot(b)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-127-9a1f5761fa9d> in <module>()
----> 1 a.dot(b)

ValueError: objects are not aligned

EDIT:

I don't understand the difference between Shape of (N,) and (N,1) and it relates to the dot() documentation.

V of shape (N,) implies an 1D array of length N, whilst shape (N, 1) implies a 2D array with N rows, 1 column:

In [2]: V = np.arange(2)

In [3]: V.shape
Out[3]: (2,)

In [4]: Q = V[:, np.newaxis]

In [5]: Q.shape
Out[5]: (2, 1)

In [6]: Q
Out[6]: 
array([[0],
       [1]])

As the docstring of np.dot says:

For 2-D arrays it is equivalent to matrix multiplication, and for 1-D arrays to inner product of vectors (without complex conjugation).

It also performs vector-matrix multiplication if one of the parameters is a vector. Say V.shape==(2,); M.shape==(2,2):

In [17]: V
Out[17]: array([0, 1])

In [18]: M
Out[18]: 
array([[2, 3],
       [4, 5]])

In [19]: np.dot(V, M)  #treats V as a 1*N 2D array
Out[19]: array([4, 5]) #note the result is a 1D array of shape (2,), not (1, 2)

In [20]: np.dot(M, V)  #treats V as a N*1 2D array
Out[20]: array([3, 5]) #result is still a 1D array of shape (2,), not (2, 1)

In [21]: Q             #a 2D array of shape (2, 1)
Out[21]: 
array([[0],
       [1]])

In [22]: np.dot(M, Q)  #matrix multiplication
Out[22]: 
array([[3],            #gets a result of shape (2, 1)
       [5]])