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