(Supplementary) NumPy Tutorial

NumPy arrays have a property called shape which tells us the number of entries of an array, in the case of a 1D arrays, and the number of rows and columns (presented as a tuple) in the case of 2D arrays.

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

assert(a.shape == (6,))
assert(b.shape == (3, 2))

So if we want to represent an m x n matrix, then we need to construct a 2D numpy array with the shame (m, n). Note that the shape of a 1D array is a 1-element tuple, so if we interpret a.shape[0] to be the number of its rows, as is the case for 2D arrays, then really a 1D array represents a column.

I find it useful to keep around functions for the number of rows and columns of NumPy arrays; it can help quite a bit with readability.

import numpy as np

a = np.array([[1, 2], [3, 4], [5, 6]])
num_of_rows = lambda a: a.shape[0]
num_of_cols = lambda a: a.shape[1]

assert(num_of_rows(a) == 3)
assert(num_of_cols(a) == 2)

Many of the operations on vectors and matrices that we discuss in this course are built into the NumPy library. For example, addition and scaling for vectors and matrices are done via the usual binary operators.

import numpy as np

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

assert(np.array_equal(a + b, np.array([[2, 3], [4, 5], [6, 7]])))
assert(np.array_equal(2 * b, np.array([[2, 2], [2, 2], [2, 2]])))

In particular, these operations are different than addition and multiplication of Python lists.

a = [[1, 2], [3, 4], [5, 6]]
b = [[1, 1], [1, 1], [1, 1]]

assert(a + b == [[1, 2], [3, 4], [5, 6], [1, 1], [1, 1], [1, 1]])
assert(2 * b == [[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]])

An Aside. Note the use of np.array_equal above for comparing two arrays for equality. This should be used cautiously; we often are not interested if two arrays are equal, but if they are sufficently close to each other. For this we can use np.allclose.

import numpy as np

x = 0.3
y = 0.1 + 0.1 + 0.1
a = np.array([x, x, x])
b = np.array([y, y, y])

assert(not np.array_equal(a, b))
assert(np.allclose(a, b))

Another important thing to note here is the * operator does not implement matrix multiplication as we have discussed it. Rather, it implements entry-wise multiplication, similar to addition. Standard matrix multiplication is done via operator @. And when applied to 1D arrays, it is the standard dot product.

import numpy as np

a = np.array([[1, 2], [3, 4]])
b = np.array([1, 1, 1, 1])

assert(np.array_equal(a * a, np.array([[1, 4], [9, 16]])))
assert(np.array_equal(a @ a, np.array([[7, 10], [15, 22]])))
assert(b @ b == 4)

The simplest and most common operation on NumPy arrays is accessing their elements. We can index them in the same way we index Python lists. In particular, we access entries in a 1D array in exactly the same way we would Python lists, but there are a couple special indexing tricks for 2D arrays that we will use frequently.

• We can access an individual row of a 2D array by indexing it as we normally would if it were a 2D Python list. The caveat is that a row is represented as a 1D array, which we have already noted represents a column. This is a feature, not a bug, because we often want to use a row as a column vector in later calculations.

  import numpy as np
  
  a = np.array([[1, 2], [3, 4]])
  r1 = a[0] # note the zero-indexing!
  assert(np.array_equal(r1, np.array([1, 2])))
  

• We can index with tuples to access individual entries, as opposed to double indexing (though both work).

  import numpy as np
  
  a = np.array([[1, 2], [3, 4]])
  
  assert(a[1, 1] == 4)
  assert(a[1][1] == 4)
  

• We can access columns using slicing, a commonly used tool for Python lists. The trick is that we can include a slice in a tuple that we use for indexing. So to get a column, we use the slide : as the first entry of the tuple we use for indexing. This says that we want the first entry of the tuple to range over all possible elements.

  import numpy as np
  
  a = np.array([[1, 2], [3, 4]])
  
  assert(np.array_equal(a[:,0], np.array([1, 3])))
  

• We can access multiple columns or multiple rows at a time either by indexing on lists or by slicing as above. The first is useful if we want to rearrange rows or columns. The second is useful if we want to grab a chunk of an array.

  import numpy as np
  
  a = np.array([[1, 2, 3], [4, 5, 6]])
  b = np.array([[4, 5, 6], [1, 2, 3]])
  c = np.array([[4, 5], [1, 2]])
  
  a[[0, 1]] = a[[1, 0]] # row swap
  
  assert(np.array_equal(a, b))
  assert(np.array_equal(b[:,:-1], c))
  

  In the second example, we're indexing b at a tuple of slices which says "all possible first values and all but the last possible second value." This will give you all of the columns except the last one.

A technique we have used several times, but which has been hidden from you behind helper functions, is stacking. We commonly need to add a row or a column to a matrix or combine to matrices into a single matrix. We can do this via np.hstack and np.vstack, which take tuples of arrays and stacks them either horizontally or vertically (per the name of the function). One important thing to keep in mind, and which will be a challenge in becoming comfortable with these functions, is that they must be applied to arrays of the same dimension, and the sides along which we are stacking must coincide. In particular, we cannot stack a 1D array on a 2D array, but we can make a 1D array into a 2D array and then stack it. Here is an example in which we horizontally stack a vector to the right of a matrix, something we often do if we want to create an augmented matrix out of a matrix equation.

import numpy as np

a = np.array([[1, 2], [3, 4]])
b = np.array([5, 6])
c = np.array([[1, 2, 5], [3, 4, 6]])

b.shape = (2, 1) # we can change the shape to make it 2D

assert(np.array_equal(np.hstack((a, b)), c))

• NumPy arrays are not copied on reassignment. This is something we just have to remember. If we want a copy of an array, we can use the function np.copy. There are other more efficient ways to manage array copying, but this will be sufficient for now.

  import numpy as np
  
  a = np.array([1, 2, 3, 4, 5])
  b = a
  c = np.copy(a)
  
  a[0] = -1
  
  assert(b[0] == -1)
  assert(c[0] == 1)
  

• NumPy arrays require that all of their entries are the same type. When you construct a NumPy array, it will perform conversions in order to make this the case. We've seen this in passing when we've set one of the entries of a matrix to a float so that every entry is converted to a float. We can use the property dtype to determine the type of entries in an array. Except in a few outstanding circumstances, we will be using arrays with floating point entries.
• NumPy array can be iterated over just like Python lists. If we want to iterate over the rows of a matrix, we just have to iterative over the elements of the array (a matrix is represented as an array of its rows). If we want to iterative over its columns we can iterative over the rows of its transpose (see below).