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#+LaTeX_CLASS: beamer
#+LaTeX_CLASS_OPTIONS: [presentation]
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#+LaTeX_CLASS: beamer
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#+LaTeX_HEADER: \usepackage[english]{babel} \usepackage{ae,aecompl}
#+LaTeX_HEADER: \usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet}
#+LaTeX_HEADER: \usepackage{listings}
#+LaTeX_HEADER: \usepackage{amsmath}
#+LaTeX_HEADER:\lstset{language=Python, basicstyle=\ttfamily\bfseries,
#+LaTeX_HEADER: commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen},
#+LaTeX_HEADER: showstringspaces=false, keywordstyle=\color{blue}\bfseries}
#+TITLE:
#+AUTHOR: FOSSEE
#+EMAIL: info@fossee.in
#+DATE:
#+DESCRIPTION:
#+KEYWORDS:
#+LANGUAGE: en
#+OPTIONS: H:3 num:nil toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
#+OPTIONS: TeX:t LaTeX:nil skip:nil d:nil todo:nil pri:nil tags:not-in-toc
*
#+begin_latex
\begin{center}
\vspace{12pt}
\textcolor{blue}{\huge Matrices}
\end{center}
\vspace{18pt}
\begin{center}
\vspace{10pt}
\includegraphics[scale=0.95]{../images/fossee-logo.png}\\
\vspace{5pt}
\scriptsize Developed by FOSSEE Team, IIT-Bombay. \\
\scriptsize Funded by National Mission on Education through ICT\\
\scriptsize MHRD,Govt. of India\\
\includegraphics[scale=0.30]{../images/iitb-logo.png}\\
\end{center}
#+end_latex
* Objectives
At the end of this tutorial, you will be able to,
- Create matrices using data.
- Create matrices from lists.
- Do basic matrix operations like addition,multiplication.
- Perform operations to find out the --
- inverse of a matrix
- determinant of a matrix
- eigen values and eigen vectors of a matrix
- norm of a matrix
- singular value decomposition of a matrix.
* Pre-requisite
Spoken tutorial on -
- Getting started with Lists.
- Getting started with Arrays.
- Accessing parts of Arrays.
* Exercise 1
- Create a two dimensional matrix ~m3~ of order (2, 4) with
elements 5, 6, 7, 8, 9, 10, 11, 12.
* Recall from ~array~
The following functions can also be used with matrices
- ~identity(n)~
- creates an identity matrix of order ~nXn~
- ~zeros((m,n))~
- creates a matrix of order ~mXn~ with 0's
- ~zeros\_like(A)~
- creates a matrix with 0's similar to the shape of matrix ~A~
- ~ones((m,n))~
- creates a matrix of order ~mXn~ with 1's
- ~ones\_like(A)~
- creates a matrix with 1's similar to the shape of matrix ~A~
* Exercise 2 : Frobenius norm \& inverse
- Find out the Frobenius norm of inverse of a ~4 X 4~ matrix.
:
The matrix is
: m5 = arange(1,17).reshape(4,4)
- Inverse of A,
-
#+begin_latex
$A^{-1} = inv(A)$
#+end_latex
- Frobenius norm is defined as,
-
#+begin_latex
$||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}$
#+end_latex
* Exercise 3 : Infinity norm
- Find the infinity norm of the matrix ~im5~
:
- Infinity norm is defined as,
#+begin_latex
$max([\sum_{i} abs(a_{i})^2])$
#+end_latex
* ~norm()~ method
- Frobenius norm
: In []: norm(im5)
- Infinity norm
: In []: norm(im5, ord=inf)
* eigen values \& eigen vectors
eigen values and eigen vectors
- eig()
Only eigen values
- eigvals()
* Singular Value Decomposition (~svd~)
#+begin_latex
$M = U \Sigma V^*$
#+end_latex
- U, an ~mXm~ unitary matrix over K.
-
#+begin_latex
$\Sigma$
#+end_latex
, an ~mXn~ diagonal matrix with non-negative real numbers on diagonal.
-
#+begin_latex
$V^*$
#+end_latex
, an ~nXn~ unitary matrix over K, denotes the conjugate transpose of V.
- SVD of matrix ~m5~ can be found out as,
: In []: svd(m5)
* Summary
In this tutorial, we have learnt to,
- Create matrices using arrays.
- Add and multiply the elements of matrix.
- Find out the inverse of a matrix,using the function ``inv()''.
- Use the function ``det()'' to find the determinant of a matrix.
- Calculate the norm of a matrix using the for loop and also using
the function ``norm()''.
- Find out the eigen vectors and eigen values of a matrix, using
functions ``eig()'' and ``eigvals()''.
- Calculate singular value decomposition(SVD) of a matrix using the
function ``svd()''.
* Evaluation
1. A and B are two array objects. Element wise multiplication in
matrices are done by,
- A * B
- multiply(A, B)
- dot(A, B)
- element\_multiply(A,B)
2. ``eig(A)[ 1 ]'' and ``eigvals(A)'' are the same.
- True
- False
3. ``norm(A,ord='fro')'' is the same as ``norm(A)'' ?
- True
- False
* Solutions
1. A * B
2. False
3. True
*
#+begin_latex
\begin{block}{}
\begin{center}
\textcolor{blue}{\Large THANK YOU!}
\end{center}
\end{block}
\begin{block}{}
\begin{center}
For more Information, visit our website\\
\url{http://fossee.in/}
\end{center}
\end{block}
#+end_latex
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