Python-3.4.3/C3/Basic-Matrix-Operations/English

Title of script: Basic Matrix Operations

Author: Puneeth, Thirumalesh H S, Arun KP

Keywords: Python, IPython, matrices, determinant, reshape, arange, eigen values, eigen vectors, transpose of matrix

Visual Cue	Narration
Show Slide title	Welcome to the spoken tutorial on Basic Matrix Operations.
Show Slide Objectives	In this tutorial, you will learn to, 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
Show Slide System Specifications	To record this tutorial, I am using Ubuntu Linux 16.04 operating system Python 3.4.3 IPython 5.1.0
Show Slide Pre-requisites	To practise this tutorial, you should have basic knowledge about Lists arrays and accessing parts of arrays and theoretical knowledge of matrix operations If not, see the relevant Python tutorials on this website.
Slide:	* In python, we create a matrix using numpy matrix class. Matrix operations can be done using numpy operators and functions.
Type, ipython3	Let us start ipython. Open the terminal.
Type ipython3 Press Enter	Type ipython3 and press Enter. From here onwards remember to press the Enter key after typing every command on the terminal.
Type, from numpy import matrix m1 = matrix([1,2,3,4]) m1 Point to the output	Let us create a matrix m1. 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.
Type, m1.shape Highlight the output	This can be verified by typing m1.shape This gives the output as (1, 4)
Type, l1 = [[1,2,3,4],[5,6,7,8]] m2 = matrix(l1) print(m2) Highlight the output	A list can also be converted to a matrix as follows, Type as shown. You can see the matrix m2 with values from list l1.
Slide:asmatrix	* 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.
Highlight according to narration Type, from numpy import asmatrix,arange m2_array = asmatrix(arange(1,9).reshape(2,4)) m2_array	Type as shown. 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.
	Pause the video. Try this exercise and then resume the video.
Show Slide Assignment 1	Create a two dimensional matrix m3 of shape 2 by 4 with the elements 5, 6, 7, 8, 9, 10, 11, 12. Hint: Use arange() and reshape() methods and asmatrix() function.
Switch to the terminal	Switch back to the terminal for the solution.
Type, m3 = asmatrix(arange(5,13).reshape(2,4)) Type, m3	Type, 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.
Type, m3 + m2	Next let us see some matrix operations. 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.
Type, m3 - m2	Similarly, type m3 minus m2 It performs matrix subtraction, that is element by element subtraction. Note that both the matrices should be of the same shape.
Type, 6.5 * m2	Now we can multiply a scalar i.e a number by a matrix as shown.
Type, m2.shape	Next we will check the size of m2 by typing, m2.shape. We get a tuple (2, 4). Matrix m2 is of the shape, two by four,
Type, m4 = asmatrix(arange(1,9).reshape(4,2))	Let us create another matrix, of the order 4 by 2. Type, m4 is equal to asmatrix inside brackets arange inside brackets 1 comma 9 dot reshape inside brackets 4 comma 2
Type m4.shape	Now to check the shape, type, m4.shape We get (4,2) as the shape of m4.
Type, m2 * m4 Highlight the output	The multiplication operator asterisk is used for matrix multiplication. Type, m2 asterisk m4 Now we get output as multiplication of m2 and m4.
Type, print (m4)	Let us now see, how to find out the transpose of a matrix. To see the content of m4, type print inside brackets m4
Type, print(m4.T) Point to the output	Now type, print inside brackets m4 dot capital T As you saw, m4 dot capital T will give the transpose of a matrix.
Show Slide:Determinant of a matrix	The determinant of a square matrix is obtained by using the function det() in numpy.linalg module.
	Pause the video. Try this exercise and resume the video.
Show Slide: Exercise	Find out the determinant of this 3 by 3 matrix.
Switch to the terminal for solution.	Switch to the terminal for the solution.
Type, from numpy.linalg import det m5 = matrix([[2,-3,1],[2,0,-1],[1,4,5]]) det(m5)	Type as shown. The determinant of m5 can be found by issuing the command, det inside brackets m5 We get determinant of m5 as output.
Show Slide Inverse of a matrix	The inverse of a square matrix can be obtained using inv() function in numpy.linalg module.
Type, from numpy.linalg import inv im5 = inv(m5) Type, im5	Let us find the inverse of the matrix m5. Type as shown. Then to see the the inverse, type im5
Type, from numpy import eye,allclose allclose(im5 * m5, asmatrix(eye(3))) Highlight eye Highlight asmatrix(eye(3))) Highlight allclose Type, eye?	Type, from numpy import eye,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. To know more about these, we will check the documentation. 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.
Show Slide eigenvectors and eigenvalues	Let us now move onto 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.
Type, 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	Let us find out the eigenvalues and eigenvectors of the matrix m6. 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.
Type, eig_value = eig(m6)[0] eig_value	To get eigenvalues type,eig underscore value is equal to eig inside brackets m6 inside square brackets 0 Then type, eig underscore value As you can see eig underscore value contains eigenvalues.
Type, eig_vector = eig(m6)[1] eig_vector	To get eigenvectors type,eig underscore vector is equal to eig inside brackets m6 inside square brackets 1 Then type, eig underscore vector eig underscore vector contains eigenvector.
Type, from numpy.linalg import eigvals eig_value1 = eigvals(m6) Then type, eig_value1 Show both the outputs	The eigenvalues can also be computed using eigvals() function. Type as shown. Then type, eig underscore value1 You can see that, eig underscore value and eig underscore value1 are same.
Show Slide Summary	This brings us to the end of this tutorial. Let us summarize. 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()
Show Slide Self assessment questions	Here are some self assessment questions for you to solve A and B are two matrix objects of appropriate sizes. Which one of the below is correct for Matrix multiplication? eig inside brackets A inside square brackets 1 and eigvals inside brackets A are the same. True or False
Show Slide 13 Solution of self assessment questions	And the answers, Matrix multiplication between A and B is done by, A asterisk B 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.
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Show Slide Acknowledgment http://spoken-tutorial.org	Spoken Tutorial Project is funded by NMEICT, MHRD, Govt. of India. For more details, visit this website.
Previous slide	This is Priya from IIT Bombay signing off. Thanks for watching.

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