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

Title of script: Advanced Matrix Operations

Author: Puneeth, Thirumalesh H S, Arun KP

Keywords: Python, IPython, array, matrices, norm, svd, video tutorial

Visual Cue	Narration
Show Slide title	Welcome to the spoken tutorial on Advanced matrix operations.
Show Slide Objectives	In this tutorial, you will learn to, find Frobenius and infinity norm of a matrix find singular value decomposition of a matrix.
Show Slide System Specifications	To record this tutorial, I am using Ubuntu Linux 16.04 operating system Python 3.4.3 and IPython 5.1.0
Show Slide Pre-requisites	To practise this tutorial, you should know about Lists, arrays and accessing parts of arrays and performing basic matrix operations If not, see the relevant Python tutorials on this website.
Show Slide flatten()	First we will see about flatten function. flatten() function returns a copy of the array, collapsed into one dimension. It can be used to convert a multidimensional matrix into a single dimension matrix
Open terminal	Let us start ipython. Open the terminal.
Type, ipython3	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 asmatrix,arange a = asmatrix(arange(1,10).reshape(3,3))	Now let us see how to create arrays Type from numpy import asmatrix,arange a is equal to asmatrix inside brackets arange inside brackets 1 comma 10 dot reshape inside brackets 3 comma 3
Type a a.flatten() Highlight numpy import asmatrix,arange	Then type, a Now type, a dot flatten open and close brackets First we imported arange function from numpy module.
Highlight (arange(1,10).reshape(3,3))	Here, we can see 3 by 3 matrix is converted into one dimensional matrix.
Show Slide Frobenius norm of a matrix	Next we will see about frobenius norm. It is defined as the square root of the sum of the absolute squares of its elements.
	Pause the video. Try this exercise and then resume the video.
Show Slide Assignment 1: Frobenius norm	Find out the frobenius norm of the inverse of the given 4 by 4 matrix.
Switch to terminal	Switch back to the terminal for the solution.
Type, m = asmatrix(arange(1,17).reshape(4,4)) Highlight asmatrix(arange(1,17).reshape(4,4))	Type m is equal to asmatrix inside brackets arange inside brackets 1 comma 17 dot reshape inside brackets 4 comma 4 Here, we have used asmatrix, arange and reshape functions. We created a matrix of size 4 by 4 containing elements from 1 to 16.
Type, m[0,1] = 0 m[1,3] =0	Now type, m inside square brackets 0 comma 1 is equal to 0 m inside square brackets 1 comma 3 is equal to 0
Type m	Then type, m We changed the value of element at row 0 column 1 and row 1 column 3 to 0.
Type, from numpy.linalg import inv, norm im = inv(m) norm(im) Highlight numpy.linalg import inv, norm	In order to find out the Frobenius norm of the inverse of matrix m, type as shown. norm function is available in numpy.linalg module.
Show Slide Infinity norm	Next, we will see about infinity norm of a matrix. It is defined as the maximum value of sum of the absolute value of elements in each row.
	Pause the video. Try this exercise and then resume the video.
Slide Assignment 2: Infinity norm	Find the infinity norm of the matrix im.
Switch to terminal	Switch back to the terminal for the solution.
Type, from numpy import infnorm(im,ord=inf)	To find out the Infinity norm of the matrix im, type as shown. Here value for ord parameter is passed as inf to calculate infinity norm.
Type, norm?	To know more about norms type norm question mark Press q to exit.
Show Slide Singular value decomposition	Next we will see about singular value decomposition. In linear algebra, the singular value decomposition is factorization of real or complex matrix.
Type, from numpy import matrix from numpy.linalg import svd m1 = matrix([[3,2,2],[2,3,-2]]) U,sigma,V_conjugate = svd(m1)	The SVD of matrix m1 can be found using svd function available in the numpy.linalg module. Type as shown. svd returns a tuple of 3 elements. We have unpacked these values into variable U, sigma and V underscore conjugate.
Type, U sigma V_conjugate	Type, Capital U Type, sigma Type, Capital V underscore conjugate
Narration only	We can validate the singular value decomposition by comparing the product of: U, sigma and V underscore conjugate with m1 sigma is a one dimensional array which contains only the diagonal elements of the matrix.
Type, from numpy import diag,allclose from numpy.matlib import zeros smat = zeros((2,3))	Type as shown. We first convert this array to a matrix.
Type, smat	Type smat smat is a 2 by 3 zero matrix
Type, smat[:2, :2] = diag(sigma)	Now type, smat inside square brackets colon 2 comma colon 2 is equal to diag inside brackets sigma
Type smat	Then type smat This replaces values at row 0 column 0 and row 1 column 1 in smat with values from sigma.
Type allclose(m1, U * smat * V_conjugate)	Type as shown. It returns True. It means elements in m1 and in product of U, sigma and V underscore conjugate are equal.
Show Slide Summary	This brings us to the end of this tutorial. Let us summarize. In this tutorial, we have learnt to, Calculate the norm of a matrix using the function norm() Calculate SVD of a matrix using the function svd()
Show Slide Self assessment questions slide.	Here is a self assessment question for you to solve 1. norm inside brackets A comma ord is equal to inside single quotes fro is the same as norm inside brackets A True or False.
Show Slide Solution of self assessment questions on slide	And the answer is True since the order is equal to inside single quotes fro stands for Frobenius norm.
Show Slide Forum	Please post your timed queries in this forum.
Show Slide Fossee Forum	Please post your general queries on Python in this forum.
Slide TBC	FOSSEE team coordinates the TBC project.
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.

Contributors and Content Editors

Nancyvarkey, Nirmala Venkat, Priyacst