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

Objectives

• find Frobenius and infinity norm of a matrix
• find singular value decomposition of a matrix.

System Specifications

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

Pre-requisites

• Lists, arrays and accessing parts of arrays and
• performing basic matrix operations

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

flatten()

• 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 the terminal.

ipython3

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

from numpy import asmatrix,arange

a = asmatrix(arange(1,10).reshape(3,3))

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

a.flatten()

Highlight numpy import asmatrix,arange

Now type, a dot flatten open and close brackets

First we imported arange function from numpy module.

Frobenius norm of a matrix

It is defined as the square root of the sum of the absolute squares of its elements.

Try this exercise and then resume the video.

Assignment 1: Frobenius norm

m = asmatrix(arange(1,17).reshape(4,4))

Highlight asmatrix(arange(1,17).reshape(4,4))

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.

m[0,1] = 0

m[1,3] =0

m inside square brackets 0 comma 1 is equal to 0

m inside square brackets 1 comma 3 is equal to 0

We changed the value of element at row 0 column 1 and row 1 column 3 to 0.

from numpy.linalg import inv, norm

im = inv(m)

norm(im)

Highlight numpy.linalg import inv, norm

norm function is available in numpy.linalg module.

Infinity norm

It is defined as the maximum value of sum of the absolute value of elements in each row.

Try this exercise and then resume the video.

from numpy import infnorm(im,ord=inf)

Here value for ord parameter is passed as inf to calculate infinity norm.

Press q to exit.

Singular value decomposition

In linear algebra, the singular value decomposition is factorization of real or complex matrix.

from numpy import matrix

from numpy.linalg import svd

m1 = matrix([[3,2,2],[2,3,-2]])

U,sigma,V_conjugate = svd(m1)

Type as shown.

svd returns a tuple of 3 elements.

We have unpacked these values into variable U, sigma and V underscore conjugate.

U

sigma

V_conjugate

Type, sigma

Type, Capital V underscore conjugate

U, sigma and V underscore conjugate with m1

sigma is a one dimensional array which contains only the diagonal elements of the matrix.

from numpy import diag,allclose

from numpy.matlib import zeros

smat = zeros((2,3))

We first convert this array to a matrix.

smat

smat is a 2 by 3 zero matrix

smat[:2, :2] = diag(sigma)

smat inside square brackets colon 2 comma colon 2 is equal to diag inside brackets sigma

This replaces values at row 0 column 0 and row 1 column 1 in smat with values from sigma.

allclose(m1, U * smat * V_conjugate)

It returns True.

It means elements in m1 and in product of U, sigma and V underscore conjugate are equal.

Summary

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()

Self assessment questions slide.

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.

Solution of self assessment questions on slide

Acknowledgment

http://spoken-tutorial.org

For more details, visit this website.

Thanks for watching.