matrices/slides.org
author anand
Mon, 08 Nov 2010 01:36:47 +0530
changeset 455 f5b7d0b693d9
parent 307 1a73dddb1d05
child 394 1a79f9ee7f5c
permissions -rw-r--r--
checklist OK
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
307
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     1
#+LaTeX_CLASS: beamer
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     2
#+LaTeX_CLASS_OPTIONS: [presentation]
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     3
#+BEAMER_FRAME_LEVEL: 1
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     4
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     5
#+BEAMER_HEADER_EXTRA: \usetheme{Warsaw}\usecolortheme{default}\useoutertheme{infolines}\setbeamercovered{transparent}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     6
#+COLUMNS: %45ITEM %10BEAMER_env(Env) %10BEAMER_envargs(Env Args) %4BEAMER_col(Col) %8BEAMER_extra(Extra)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     7
#+PROPERTY: BEAMER_col_ALL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 :ETC
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     8
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
     9
#+LaTeX_CLASS: beamer
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    10
#+LaTeX_CLASS_OPTIONS: [presentation]
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    11
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    12
#+LaTeX_HEADER: \usepackage[english]{babel} \usepackage{ae,aecompl}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    13
#+LaTeX_HEADER: \usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    14
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    15
#+LaTeX_HEADER: \usepackage{listings}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    16
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    17
#+LaTeX_HEADER:\lstset{language=Python, basicstyle=\ttfamily\bfseries,
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    18
#+LaTeX_HEADER:  commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen},
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    19
#+LaTeX_HEADER:  showstringspaces=false, keywordstyle=\color{blue}\bfseries}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    20
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    21
#+TITLE: Matrices
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    22
#+AUTHOR: FOSSEE
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    23
#+EMAIL:     
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    24
#+DATE:    
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    25
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    26
#+DESCRIPTION: 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    27
#+KEYWORDS: 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    28
#+LANGUAGE:  en
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    29
#+OPTIONS:   H:3 num:nil toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    30
#+OPTIONS:   TeX:t LaTeX:nil skip:nil d:nil todo:nil pri:nil tags:not-in-toc
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    31
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    32
* Outline
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    33
  - Creating Matrices
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    34
    - using direct data
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    35
    - converting a list
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    36
  - Matrix operations
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    37
  - Inverse of matrix
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    38
  - Determinant of matrix
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    39
  - Eigen values and Eigen vectors of matrices
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    40
  - Norm of matrix
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    41
  - Singular Value Decomposition of matrices
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    42
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    43
* Creating a matrix
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    44
  - Creating a matrix using direct data
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    45
  : In []: m1 = matrix([1, 2, 3, 4])
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    46
  - Creating a matrix using lists
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    47
  : In []: l1 = [[1,2,3,4],[5,6,7,8]]
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    48
  : In []: m2 = matrix(l1)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    49
  - A matrix is basically an array
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    50
  : In []: m3 = array([[5,6,7,8],[9,10,11,12]])
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    51
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    52
* Matrix operations
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    53
  - Element-wise addition (both matrix should be of order ~mXn~)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    54
    : In []: m3 + m2
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    55
  - Element-wise subtraction (both matrix should be of order ~mXn~)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    56
    : In []: m3 - m2
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    57
* Matrix Multiplication
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    58
  - Matrix Multiplication
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    59
    : In []: m3 * m2
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    60
    : Out []: ValueError: objects are not aligned
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    61
  - Element-wise multiplication using ~multiply()~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    62
    : multiply(m3, m2)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    63
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    64
* Matrix Multiplication (cont'd)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    65
  - Create two compatible matrices of order ~nXm~ and ~mXr~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    66
    : In []: m1.shape
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    67
    - matrix m1 is of order ~1 X 4~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    68
  - Creating another matrix of order ~4 X 2~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    69
    : In []: m4 = matrix([[1,2],[3,4],[5,6],[7,8]])
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    70
  - Matrix multiplication
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    71
    : In []: m1 * m4
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    72
* Recall from ~array~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    73
  - The functions 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    74
    - ~identity(n)~ - 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    75
      creates an identity matrix of order ~nXn~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    76
    - ~zeros((m,n))~ - 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    77
      creates a matrix of order ~mXn~ with 0's
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    78
    - ~zeros_like(A)~ - 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    79
      creates a matrix with 0's similar to the shape of matrix ~A~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    80
    - ~ones((m,n))~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    81
      creates a matrix of order ~mXn~ with 1's
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    82
    - ~ones_like(A)~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    83
      creates a matrix with 1's similar to the shape of matrix ~A~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    84
  Can also be used with matrices
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    85
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    86
* More matrix operations
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    87
  Transpose of a matrix
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    88
  : In []: m4.T
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    89
* Exercise 1 : Frobenius norm \& inverse
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    90
  Find out the Frobenius norm of inverse of a ~4 X 4~ matrix.
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    91
  : 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    92
  The matrix is
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    93
  : m5 = matrix(arange(1,17).reshape(4,4))
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    94
  - Inverse of A, 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    95
    - 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    96
     #+begin_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    97
       $A^{-1} = inv(A)$
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    98
     #+end_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
    99
  - Frobenius norm is defined as,
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   100
    - 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   101
      #+begin_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   102
        $||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}$
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   103
      #+end_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   104
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   105
* Exercise 2: Infinity norm
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   106
  Find the infinity norm of the matrix ~im5~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   107
  : 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   108
  - Infinity norm is defined as,
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   109
    #+begin_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   110
       $max([\sum_{i} abs(a_{i})^2])$
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   111
    #+end_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   112
* ~norm()~ method
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   113
  - Frobenius norm
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   114
    : In []: norm(im5)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   115
  - Infinity norm
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   116
    : In []: norm(im5, ord=inf)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   117
* Determinant
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   118
  Find out the determinant of the matrix m5
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   119
  : 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   120
  - determinant can be found out using
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   121
    - ~det(A)~ - returns the determinant of matrix ~A~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   122
* eigen values \& eigen vectors
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   123
  Find out the eigen values and eigen vectors of the matrix ~m5~.
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   124
  : 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   125
  - eigen values and vectors can be found out using
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   126
    : In []: eig(m5)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   127
    returns a tuple of /eigen values/ and /eigen vectors/
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   128
  - /eigen values/ in tuple
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   129
    - ~In []: eig(m5)[0]~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   130
  - /eigen vectors/ in tuple
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   131
    - ~In []: eig(m5)[1]~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   132
  - Computing /eigen values/ using ~eigvals()~
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   133
    : In []: eigvals(m5)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   134
* Singular Value Decomposition (~svd~)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   135
  #+begin_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   136
    $M = U \Sigma V^*$
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   137
  #+end_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   138
    - U, an ~mXm~ unitary matrix over K.
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   139
    - 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   140
      #+begin_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   141
        $\Sigma$
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   142
      #+end_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   143
	, an ~mXn~ diagonal matrix with non-negative real numbers on diagonal.
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   144
    - 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   145
      #+begin_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   146
        $V^*$
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   147
      #+end_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   148
	, an ~nXn~ unitary matrix over K, denotes the conjugate transpose of V.
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   149
  - SVD of matrix ~m5~ can be found out as,
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   150
    : In []: svd(m5)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   151
* Summary
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   152
  - Matrices
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   153
    - creating matrices
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   154
  - Matrix operations
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   155
  - Inverse (~inv()~)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   156
  - Determinant (~det()~)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   157
  - Norm (~norm()~)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   158
  - Eigen values \& vectors (~eig(), eigvals()~)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   159
  - Singular Value Decomposition (~svd()~)
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   160
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   161
* Thank you!
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   162
#+begin_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   163
  \begin{block}{}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   164
  \begin{center}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   165
  This spoken tutorial has been produced by the
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   166
  \textcolor{blue}{FOSSEE} team, which is funded by the 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   167
  \end{center}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   168
  \begin{center}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   169
    \textcolor{blue}{National Mission on Education through \\
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   170
      Information \& Communication Technology \\ 
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   171
      MHRD, Govt. of India}.
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   172
  \end{center}  
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   173
  \end{block}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   174
#+end_latex
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   175
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents:
diff changeset
   176