Understand Tensors With Numpy

In this notebook, I will explain tensors with Python Numpy. For numpy refresher please checkout following resource...
https://www.nbshare.io/notebook/692176011/Numpy-Basics/

In [1]:

import numpy as np

Before we delve in to tensors. We need to understand first what is Scalars, Vectors, Arrays and Matrices.

Scalar - Scalar is just a number. Example number 55 is a scalar. Scalar has no dimensions.
Vector - Vector is one dimensional array or scalar with orientation (i.e. horizontal or vertical). Vector can be 1,2 or n dimensional.
Array - Array is a computer language term used in data structures. Array is a collection of similar data elements.
Matrix - Matrix is a mathematical term. It is a 2D array of numbers represented in rows and columns.

What is Vector

Example of 3 dimensional Vector is shown below.

In [2]:

threeDVector = np.array([1,2,3])

In [3]:

threeDVector

Out[3]:

array([1, 2, 3])

In [4]:

threeDVector.shape

Out[4]:

(3,)

What is Matrix

Matrix is 2 dimensional array.

In [5]:

np.array([[1,2,3],[4,5,6]])

Out[5]:

array([[1, 2, 3],
       [4, 5, 6]])

What is Tensor

Tensor is a mathematical object with more than two dimensions. The number of dimensions of an array defines the order of a Tensor. For example a scalar is 0 order tensor. Similarly vector is a 1 order tensor. A 2D matrix is 2nd order Tensor so on and so forth.

Below is an example of vector which is 1 order tensor.

In [6]:

np.array([1,2,3])

Out[6]:

array([1, 2, 3])

Let us define a two dimensional tensor.

In [7]:

twodTensor = np.arange(36).reshape(6,6)

In [8]:

twodTensor

Out[8]:

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],
       [24, 25, 26, 27, 28, 29],
       [30, 31, 32, 33, 34, 35]])

Let us access the ist row of above tensor

In [9]:

twodTensor[0]

Out[9]:

array([0, 1, 2, 3, 4, 5])

In [10]:

twodTensor.shape

Out[10]:

(6, 6)

Three dimensional Tensor example is shown below...

In [11]:

threedTensor = np.arange(36).reshape(4,3,3)

In [12]:

threedTensor

Out[12]:

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],
        [24, 25, 26]],

       [[27, 28, 29],
        [30, 31, 32],
        [33, 34, 35]]])

Note the 3 enclosing brackets - array[[[]]] which indicates it is 3 dimensional tensor.

Add new dimension or axis to a tensor.

In [13]:

t3t = np.arange(36).reshape(4,3,3)

Let us make our 3 dimension tensor to a 4 dimension tensor.

In [14]:

t4t = t3t[np.newaxis,:,:]

Note the 4 enclosing brackets in the below array.

In [15]:

t4t

Out[15]:

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],
         [24, 25, 26]],

        [[27, 28, 29],
         [30, 31, 32],
         [33, 34, 35]]]])

Note our 4th dimension has only 1 element (the 3x3 array). Let us see that using tensor.shape

In [16]:

t4t.shape

Out[16]:

(1, 4, 3, 3)

In [17]:

t4t[0] #4th dimension

Out[17]:

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],
        [24, 25, 26]],

       [[27, 28, 29],
        [30, 31, 32],
        [33, 34, 35]]])

Arithmetic with Tensors

Tensors Arithemetic is mathematical operations on vector and matrix.

Example - Find the dot product of two 3 dimensional vectors, we can use np.dot

In [18]:

array1 = np.array([4,4,3])
array2 = np.array([2,2,3])

In [19]:

array1.shape

Out[19]:

(3,)

In [20]:

array2.shape

Out[20]:

(3,)

In [21]:

np.dot(array1,array2)

Out[21]:

Dot product two matrix is shown below.

In [22]:

array1 = np.array([[1,2,3],[4,4,3]])
array2 = np.array([[2,2,3],[1,2,3]])

In [23]:

print(array1)
print("\n")
print(array2)

[[1 2 3]
 [4 4 3]]


[[2 2 3]
 [1 2 3]]

In [24]:

np.dot(array1,array2) #this will throw error because row of first matrix is not equal to column of second matrix - Refer matrix multiplication fundamentals

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Input In [24], in <cell line: 1>()
----> 1 np.dot(array1,array2)

File <__array_function__ internals>:180, in dot(*args, **kwargs)

ValueError: shapes (2,3) and (2,3) not aligned: 3 (dim 1) != 2 (dim 0)

In [25]:

array1 = np.array([[1,2,3],[4,4,3],[1,1,1]])
array2 = np.array([[2,2,3],[1,2,3],[1,2,3]])

Dot product for above matrices will work because both the arrays are 3x3 matrices.

In [26]:

np.dot(array1,array2)

Out[26]: