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The NumPy library is very helpful for solving linear systems in Python. For instructions on how to install NumPy look here.

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Matrices in Python

Info
Like many other programming languages, the indices in Python arrays starts at 0. This is commonly known, but may catch you by surprise if you're used to the indices in e.g. Matlab.

In Python, matrices are not it's own thing, but rather a list of list/nested lists. Let's look at an example:

Our matrix of choice will be

To represent this matrix in python, we will consider the matrix as two independent lists,

and

, put together in one list. Written in code, it looks like this:

Code Block

language	py

mat1 = [[1, 2, 3], [4, 5, 6]]
# To make the matrices easier to read while coding, we can place the different lists vertical to eachother instead of horizontally.
mat2 = [[1, 2, 3],
	   [4, 5, 6]]

# Printing both of these matrices will result in the same output:
[[1, 2, 3], [4, 5, 6]]

# To alter the output to a more readable format, for-loops are very helpful:
[[1, 2, 3], 
[4, 5, 6]]

Now, let's see how to obtain different values from our matrix:

Code Block

language	py

A = [[1, 2, 3], 
	[4, 5, 6]]

print("A =", A) 				# Whole matrix
print("A[1] =", A[1])      		# 2nd row
print("A[1][2] =", A[1][2])   	# 3rd element of 2nd row
print("A[0][-1] =", A[0][-1])   # Last element of 1st Row

column = [];        			# Empty list
for row in A:
  column.append(row[2])   		# Adding the element in the third column of every row.

print("3rd column =", column)

The script above will result in the following output:

Code Block

language	text

A = [[1, 2, 3], [4, 5, 6]]
A[1] = [4, 5, 6]
A[1][2] = 6
A[0][-1] = 3
3rd column = [3, 6]

Matrices in Python using NumPy

First of all, we have to import the NumPy libary. For the sake of this tutorial, we will only do it once, but you can assume it is done everywhere NumPy is being used.

Info
A lot of great, more detailed information about the different functions in NumPy can be found by searching in their user manual. This article will only describe a limited number of functions and their functionalities.

Code Block

language	py

import numpy as np

To make an array (or matrix) using NumPy, we will use the function numpy.array, and simply use the same syntax as before, but now as a function parameter.

Note

There is another NumPy function for making matrices, numpy.matrix, but this is no longer recommended to use. The class may be removed in the future.

BibTeX Referencing

reference	@misc{scipy, title={numpy.matrix¶}, url={https://docs.scipy.org/doc/numpy/reference/generated/numpy.matrix.html?highlight=matrix#numpy.matrix}, journal={numpy.matrix - NumPy v1.16 Manual}, author={SciPy}}

Code Block

language	py

A = np.array([[1, 2, 3], [4, 5, 6]])		# Array of ints


B = np.array([[1.1, 2, 3], [4, 5, 6]])		# Array of floats


C = np.array([[1, 2, 3], [4, 5, 6]], dtype=complex)		# Array of complex numbers

When printing the matrices above using print, the output will look like this:

Code Block

language	py

[[1 2 3]					# A
 [4 5 6]]

[[1.1 2.  3. ]				# B
 [4.  5.  6. ]]

[[1.+0.j 2.+0.j 3.+0.j]		# C
 [4.+0.j 5.+0.j 6.+0.j]]

Notice how the matrices are printed in an easily readable format without the use of for-loops. This is another benefit of using NumPy arrays.

There are several other functions in NumPy that can create specific matrices. Here are some of them:

numpy.zeros returns a new array with given shape and type, filled with zeros.
Code Block
language py
title zeros
collapse true
A = np.zeros((2, 3)) # When printed: [[0. 0. 0.] [0. 0. 0.]]

numpy.ones returns a new array with given shape and type, filled with ones.
Code Block
language py
title ones
collapse true
A = np.ones((2, 3)) # When printed: [[1. 1. 1.] [1. 1. 1.]]

numpy.empty returns a new array with given shape and type, without initalizing entries.

Code Block

language	py
title	empty
collapse	true

A = np.empty((2, 3))

# When printed:

[[ 1.33360294e+241  1.00200641e-310  3.42196482e-210]		# Random numbers
 [ 2.42869050e-313  1.59166660e-308 -1.95565444e-310]]

numpy.full returns a new array with given shape and type, filled with fill_value.

Code Block

language	py
title	full
collapse	true

A = np.full((2, 3), 2019)

B = np.full((2, 3), np.inf)

# When printed:

[[2019 2019 2019]		# A
 [2019 2019 2019]]	

[[inf inf inf]			# B
 [inf inf inf]]

Linear algebra using NumPy

NumPy has several useful functions for linear algebra built-in. For full list look check the NumPy documentation.

Code Block

language	py
title	Solve()

np.linalg.solve(a,b) #where a and b is matrixs


#returns a array with solutions to the system

Code Block

language	py
title	Eigenvalues and eigenvectors

np.linalg.eig(a) # where a is a square matrix


#returns two arrays [v,w]
#v containing the eigenvalues of a
#w containing the eigenvectors of a

Code Block

language	py
title	Determinant

np.linalg.det(a) # where a is a square matrix


#returns the determinant of the matrix a

Code Block

language	py
title	Finding the inverse

np.linalg.inv(a) # where a is the matrix to be inverted


#Returns the inverse matrix of a

Example

Code Block

language	py
title	Solving set of equations

import numpy as np

# Solving following system of linear equation
# 5a + 2b = 35
# 1a + 4b = 49

a = np.array([[5, 2],[1,4]]) # Lefthand-side of the equation
b = np.array([35, 94]) #Righthand-side

print(np.linalg.solve(a,b)) #Printing the solution

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Matrices in Python

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