Difference between revisions of "Python-3.4.3/C3/Basic-Matrix-Operations/English"

Objectives

• Create matrices from lists
• Perform basic matrix operations like
  • addition
  • subtraction and
  • multiplication
• Perform operations to find out
  • determinant of a matrix
  • inverse of a matrix
  • eigenvalues and eigenvectors of a matrix

System Specifications

• Ubuntu Linux 16.04 operating system
• Python 3.4.3
• IPython 5.1.0

Pre-requisites

• Lists
• arrays and accessing parts of arrays and
• theoretical knowledge of matrix operations

If not, see the relevant Python tutorials on this website.

• In Python, we create a matrix using numpy matrix class.
• Matrix operations can be done using numpy operators and functions.

ipython3

Open the terminal.

Press Enter

From here onwards, remember to press the Enter key after typing every command on the terminal.

from numpy import matrix

m1 = matrix([1,2,3,4])

m1

Point to the output

Type from numpy import matrix

Then type,

m1 is equal to matrix inside brackets inside square brackets 1 comma 2 comma 3 comma 4

Now type m1

This creates a matrix with one row and four columns.

m1.shape

Highlight the output

This gives the output as (1, 4)

l1 = [[1,2,3,4],[5,6,7,8]]

m2 = matrix(l1)

print(m2)

Highlight the output

Type as shown.

You can see the matrix m2 with values from list l1.

• To convert an array to a matrix, use the asmatrix method in numpy module.
• We can use arange and reshape methods to generate an array.

Type,

from numpy import asmatrix,arange

m2_array = asmatrix(arange(1,9).reshape(2,4))

m2_array

arange is a method available in numpy.

Here it returns an array of evenly spaced values between 1 and 9.

reshape is used to change the shape of the array to 2 rows and 4 columns.

asmatrix is a method available in numpy and it interprets the input as a matrix.

Try this exercise and then resume the video.

Assignment 1

Hint: Use arange() and reshape() methods and asmatrix() function.

m3 = asmatrix(arange(5,13).reshape(2,4))

Type, m3

m3 is equal to asmatrix inside brackets arange inside brackets 5 comma 13 dot reshape inside brackets 2 comma 4

Type, m3

You can see the required output.

m3 + m2

Type, m3 plus m2

It performs element by element addition, that is matrix addition.

Note that both the matrices should be of the same shape.

m3 - m2

It performs matrix subtraction, that is element by element subtraction.

Note that both the matrices should be of the same shape.

6.5 * m2

m2.shape

m2.shape.

We get a tuple (2, 4).

Matrix m2 is of the shape, two by four,

m4 = asmatrix(arange(1,9).reshape(4,2))

Type,

m4 is equal to asmatrix inside brackets arange inside brackets 1 comma 9 dot reshape inside brackets 4 comma 2

m4.shape

We get (4,2) as the shape of m4.

m2 * m4

Highlight the output

Type m2 asterisk m4

Now we get output as multiplication of m2 and m4.

print (m4)

To see the content of m4, type print inside brackets m4

print(m4.T)

Point to the output

print inside brackets m4 dot capital T

As you saw, m4 dot capital T will give the transpose of a matrix.

Try this exercise and resume the video.

from numpy.linalg import det

m5 = matrix([[2,-3,1],[2,0,-1],[1,4,5]])

det(m5)

The determinant of m5 can be found by issuing the command,

det inside brackets m5

We get determinant of m5 as output.

Inverse of a matrix

from numpy.linalg import inv

im5 = inv(m5)

Type,

im5

Type as shown.

Then to see the the inverse, type

im5

from numpy import eye,allclose

allclose(im5 * m5, asmatrix(eye(3)))

Highlight eye

Highlight asmatrix(eye(3)))

Highlight allclose

Then type,

allclose inside brackets im5 asterisk m5 comma asmatrix inside brackets eye inside brackets 3

This returns True.

We know that multiplication of a matrix with its inverse gives the identity matrix.

Identity matrix is created using eye() function. It is present in the numpy module.

Here asmatrix inside brackets eye inside brackets 3 gives identity matrix of size 3.

allclose is a function that returns True if two arrays are element-wise equal.

eye?

Type the function name followed by a question mark in IPython console.

Type eye question mark

To quit the documentation, press q.

It is a good practice to read documentation of new functions that you come across.

eigenvectors and eigenvalues

Given a square matrix A

• eig inside brackets A inside square brackets 0 gives its eigenvalues
• eig inside brackets A inside square brackets 1 gives its eigenvector
• eigvals inside brackets A gives its eigenvalues

eig and eigvals functions are present in numpy.linalg module.

from numpy import diag

from numpy.linalg import eig

m6=asmatrix(diag((1, 2, 3)))

Type,

eig(m6)

Highlight diag((1, 2, 3)))

Highlight

(array([1., 2., 3.]), matrix([[1., 0., 0.],

[0., 1., 0.],

[0., 0., 1.]]))

Put box to array

Put box to matrix

Type as shown.

Now to see the value, type,eig inside brackets m6

diag inside brackets again inside brackets 1 comma 2 comma 3 creates a diagonal matrix with 1,2,3 as diagonal elements and 0 elsewhere .

diag() function is present in numpy module.

Note that eig inside brackets m6 returned a tuple of one array and one matrix.

The first element in the tuple is an array of three eigenvalues.

The second element in the tuple is a matrix of three eigenvectors.

eig_value = eig(m6)[0]

eig_value

Then type eig underscore value

As you can see eig underscore value contains eigenvalues.

eig_vector = eig(m6)[1]

eig_vector

Then type eig underscore vector

eig underscore vector contains eigenvector.

from numpy.linalg import eigvals

eig_value1 = eigvals(m6)

Then type eig_value1

Show both the outputs.

Type as shown.

Then type eig underscore value1

You can see that, eig underscore value and eig underscore value1 are same.

Summary

In this tutorial, we have learnt to,

• Create matrices using arrays
• Add, subtract and multiply matrices
• Take scalar multiple of a matrix
• Use the function det() to find the determinant of a matrix
• Find out the inverse of a matrix using the function inv()
• Find out the eigenvectors and eigenvalues of a matrix, using the functions eig() and eigvals()

Self assessment questions

1. A and B are two matrix objects of appropriate sizes. Which one of the below is correct for matrix multiplication?
2. eig inside brackets A inside square brackets 1 and eigvals inside brackets A are the same. True or False?

Solution of self assessment questions

1. Matrix multiplication between A and B is done by, A asterisk B
2. False. eig inside brackets A inside square brackets 0 and eigvals inside brackets A are same, that is both will give the eigenvalues of matrix A.

Acknowledgment

http://spoken-tutorial.org

For more details, visit this website.

Thanks for watching.