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#+TITLE:  
#+AUTHOR:  FOSSEE
#+EMAIL:   info@fossee.in
#+DATE:    

#+DESCRIPTION: 
#+KEYWORDS: 
#+LANGUAGE:  en
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* 
#+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