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matrices/slides.tex
changeset 394 1a79f9ee7f5c
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     1 % Created 2010-10-12 Tue 14:28
 
     1 % Created 2010-11-07 Sun 16:18
 
     2 \documentclass[presentation]{beamer}
 
     2 \documentclass[presentation]{beamer}
 
     3 \usepackage[latin1]{inputenc}
 
     3 \usepackage[latin1]{inputenc}
 
     4 \usepackage[T1]{fontenc}
 
     4 \usepackage[T1]{fontenc}
 
     5 \usepackage{fixltx2e}
 
     5 \usepackage{fixltx2e}
 
     6 \usepackage{graphicx}
 
     6 \usepackage{graphicx}
 
    68 \begin{itemize}
 
    68 \begin{itemize}
 
    69 \item Creating a matrix using direct data
 
    69 \item Creating a matrix using direct data
 
    70 \end{itemize}
 
    70 \end{itemize}
 
    71 
    71 
    72 \begin{verbatim}
 
    72 \begin{verbatim}
 
    73    In []: m1 = matrix([1, 2, 3, 4])
 
    73    In []: m1 = array([1, 2, 3, 4])
 
    74 \end{verbatim}
 
    74 \end{verbatim}
 
    75 
    75 
    76 \begin{itemize}
 
    76 \begin{itemize}
 
    77 \item Creating a matrix using lists
 
    77 \item Creating a matrix using lists
 
    78 \end{itemize}
 
    78 \end{itemize}
 
    79 
    79 
    80 \begin{verbatim}
 
    80 \begin{verbatim}
 
    81    In []: l1 = [[1,2,3,4],[5,6,7,8]]
 
    81    In []: l1 = [[1,2,3,4],[5,6,7,8]]
 
    82    In []: m2 = matrix(l1)
 
    82    In []: m2 = array(l1)
 
    83 \end{verbatim}
 
    83 \end{verbatim}
 
    84 
    84 \end{frame}
 
    85 \begin{itemize}
 
    85 \begin{frame}[fragile]
 
    86 \item A matrix is basically an array
 
    86 \frametitle{Exercise 1}
 
       
    87 \label{sec-3}
 
       
    88 
       
    89   Create a (2, 4) matrix \texttt{m3}
 
       
    90 \begin{verbatim}
 
       
    91    m3 = [[5,  6,  7,  8],
 
       
    92          [9, 10, 11, 12]]
 
       
    93 \end{verbatim}
 
       
    94 \end{frame}
 
       
    95 \begin{frame}[fragile]
 
       
    96 \frametitle{Solution 1}
 
       
    97 \label{sec-4}
 
       
    98 
       
    99 \begin{itemize}
 
       
   100 \item m3 can be created as,
 
    87 \end{itemize}
 
   101 \end{itemize}
 
    88 
   102 
    89 \begin{verbatim}
 
   103 \begin{verbatim}
 
    90    In []: m3 = array([[5,6,7,8],[9,10,11,12]])
 
   104    In []: m3 = array([[5,6,7,8],[9,10,11,12]])
 
    91 \end{verbatim}
 
   105 \end{verbatim}
 
    92 \end{frame}
 
   106 \end{frame}
 
    93 \begin{frame}[fragile]
 
   107 \begin{frame}[fragile]
 
    94 \frametitle{Matrix operations}
 
   108 \frametitle{Matrix operations}
 
    95 \label{sec-3}
 
   109 \label{sec-5}
 
    96 
   110 
    97 \begin{itemize}
 
   111 \begin{itemize}
 
    98 \item Element-wise addition (both matrix should be of order \texttt{mXn})
 
   112 \item Element-wise addition (both matrix should be of order \texttt{mXn})
 
    99 \begin{verbatim}
 
   113 \begin{verbatim}
 
   100      In []: m3 + m2
 
   114      In []: m3 + m2
 
   107 
   121 
   108 \end{itemize}
 
   122 \end{itemize}
 
   109 \end{frame}
 
   123 \end{frame}
 
   110 \begin{frame}[fragile]
 
   124 \begin{frame}[fragile]
 
   111 \frametitle{Matrix Multiplication}
 
   125 \frametitle{Matrix Multiplication}
 
   112 \label{sec-4}
 
   126 \label{sec-6}
 
   113 
   127 
   114 \begin{itemize}
 
   128 \begin{itemize}
 
   115 \item Matrix Multiplication
 
   129 \item Element-wise multiplication using \texttt{m3 * m2}
 
   116 \begin{verbatim}
 
   130 \begin{verbatim}
 
   117      In []: m3 * m2
 
   131      In []: m3 * m2
 
       
   132 \end{verbatim}
 
       
   133 
       
   134 \item Matrix Multiplication using \texttt{dot(m3, m2)}
 
       
   135 \begin{verbatim}
 
       
   136      In []: dot(m3, m2)
 
   118      Out []: ValueError: objects are not aligned
 
   137      Out []: ValueError: objects are not aligned
 
   119 \end{verbatim}
 
   138 \end{verbatim}
 
   120 
   139 
   121 \item Element-wise multiplication using \texttt{multiply()}
 
       
   122 \begin{verbatim}
 
       
   123      multiply(m3, m2)
 
       
   124 \end{verbatim}
 
       
   125 
       
   126 \end{itemize}
 
   140 \end{itemize}
 
   127 \end{frame}
 
   141 \end{frame}
 
   128 \begin{frame}[fragile]
 
   142 \begin{frame}[fragile]
 
   129 \frametitle{Matrix Multiplication (cont'd)}
 
   143 \frametitle{Matrix Multiplication (cont'd)}
 
   130 \label{sec-5}
 
   144 \label{sec-7}
 
   131 
   145 
   132 \begin{itemize}
 
   146 \begin{itemize}
 
   133 \item Create two compatible matrices of order \texttt{nXm} and \texttt{mXr}
 
   147 \item Create two compatible matrices of order \texttt{nXm} and \texttt{mXr}
 
   134 \begin{verbatim}
 
   148 \begin{verbatim}
 
   135      In []: m1.shape
 
   149      In []: m1.shape
 
   140 \item matrix m1 is of order \texttt{1 X 4}
 
   154 \item matrix m1 is of order \texttt{1 X 4}
 
   141 \end{itemize}
 
   155 \end{itemize}
 
   142 
   156 
   143 \item Creating another matrix of order \texttt{4 X 2}
 
   157 \item Creating another matrix of order \texttt{4 X 2}
 
   144 \begin{verbatim}
 
   158 \begin{verbatim}
 
   145      In []: m4 = matrix([[1,2],[3,4],[5,6],[7,8]])
 
   159      In []: m4 = array([[1,2],[3,4],[5,6],[7,8]])
 
   146 \end{verbatim}
 
   160 \end{verbatim}
 
   147 
   161 
   148 \item Matrix multiplication
 
   162 \item Matrix multiplication
 
   149 \begin{verbatim}
 
   163 \begin{verbatim}
 
   150      In []: m1 * m4
 
   164      In []: dot(m1, m4)
 
   151 \end{verbatim}
 
   165 \end{verbatim}
 
   152 
   166 
   153 \end{itemize}
 
   167 \end{itemize}
 
   154 \end{frame}
 
   168 \end{frame}
 
   155 \begin{frame}
 
   169 \begin{frame}
 
   156 \frametitle{Recall from \texttt{array}}
 
   170 \frametitle{Recall from \texttt{array}}
 
   157 \label{sec-6}
 
   171 \label{sec-8}
 
   158 
   172 
   159 \begin{itemize}
 
   173 \begin{itemize}
 
   160 \item The functions
 
   174 \item The functions
 
   161 
   175 
   162 \begin{itemize}
 
   176 \begin{itemize}
 
   176 
   190 
   177   Can also be used with matrices
 
   191   Can also be used with matrices
 
   178 \end{frame}
 
   192 \end{frame}
 
   179 \begin{frame}[fragile]
 
   193 \begin{frame}[fragile]
 
   180 \frametitle{More matrix operations}
 
   194 \frametitle{More matrix operations}
 
   181 \label{sec-7}
 
   195 \label{sec-9}
 
   182 
   196 
   183   Transpose of a matrix
 
   197   Transpose of a matrix
 
   184 \begin{verbatim}
 
   198 \begin{verbatim}
 
   185    In []: m4.T
 
   199    In []: m4.T
 
   186 \end{verbatim}
 
   200 \end{verbatim}
 
   187 \end{frame}
 
   201 \end{frame}
 
   188 \begin{frame}[fragile]
 
   202 \begin{frame}[fragile]
 
   189 \frametitle{Exercise 1 : Frobenius norm \& inverse}
 
   203 \frametitle{Exercise 2 : Frobenius norm \& inverse}
 
   190 \label{sec-8}
 
   204 \label{sec-10}
 
   191 
   205 
   192   Find out the Frobenius norm of inverse of a \texttt{4 X 4} matrix.
 
   206   Find out the Frobenius norm of inverse of a \texttt{4 X 4} matrix.
 
   193 \begin{verbatim}
 
   207 \begin{verbatim}
 
   194    
   208    
   195 \end{verbatim}
 
   209 \end{verbatim}
 
   196 
   210 
   197   The matrix is
 
   211   The matrix is
 
   198 \begin{verbatim}
 
   212 \begin{verbatim}
 
   199    m5 = matrix(arange(1,17).reshape(4,4))
 
   213    m5 = arange(1,17).reshape(4,4)
 
   200 \end{verbatim}
 
   214 \end{verbatim}
 
   201 
   215 
   202 \begin{itemize}
 
   216 \begin{itemize}
 
   203 \item Inverse of A,
 
   217 \item Inverse of A,
 
   204 
   218 
   213 \end{itemize}
 
   227 \end{itemize}
 
   214 
   228 
   215 \end{itemize}
 
   229 \end{itemize}
 
   216 \end{frame}
 
   230 \end{frame}
 
   217 \begin{frame}[fragile]
 
   231 \begin{frame}[fragile]
 
   218 \frametitle{Exercise 2: Infinity norm}
 
   232 \frametitle{Exercise 3 : Infinity norm}
 
   219 \label{sec-9}
 
   233 \label{sec-11}
 
   220 
   234 
   221   Find the infinity norm of the matrix \texttt{im5}
 
   235   Find the infinity norm of the matrix \texttt{im5}
 
   222 \begin{verbatim}
 
   236 \begin{verbatim}
 
   223    
   237    
   224 \end{verbatim}
 
   238 \end{verbatim}
 
   228        $max([\sum_{i} abs(a_{i})^2])$
 
   242        $max([\sum_{i} abs(a_{i})^2])$
 
   229 \end{itemize}
 
   243 \end{itemize}
 
   230 \end{frame}
 
   244 \end{frame}
 
   231 \begin{frame}[fragile]
 
   245 \begin{frame}[fragile]
 
   232 \frametitle{\texttt{norm()} method}
 
   246 \frametitle{\texttt{norm()} method}
 
   233 \label{sec-10}
 
   247 \label{sec-12}
 
   234 
   248 
   235 \begin{itemize}
 
   249 \begin{itemize}
 
   236 \item Frobenius norm
 
   250 \item Frobenius norm
 
   237 \begin{verbatim}
 
   251 \begin{verbatim}
 
   238      In []: norm(im5)
 
   252      In []: norm(im5)
 
   245 
   259 
   246 \end{itemize}
 
   260 \end{itemize}
 
   247 \end{frame}
 
   261 \end{frame}
 
   248 \begin{frame}[fragile]
 
   262 \begin{frame}[fragile]
 
   249 \frametitle{Determinant}
 
   263 \frametitle{Determinant}
 
   250 \label{sec-11}
 
   264 \label{sec-13}
 
   251 
   265 
   252   Find out the determinant of the matrix m5
 
   266   Find out the determinant of the matrix m5
 
   253 \begin{verbatim}
 
   267 \begin{verbatim}
 
   254    
   268    
   255 \end{verbatim}
 
   269 \end{verbatim}
 
   263 
   277 
   264 \end{itemize}
 
   278 \end{itemize}
 
   265 \end{frame}
 
   279 \end{frame}
 
   266 \begin{frame}[fragile]
 
   280 \begin{frame}[fragile]
 
   267 \frametitle{eigen values \& eigen vectors}
 
   281 \frametitle{eigen values \& eigen vectors}
 
   268 \label{sec-12}
 
   282 \label{sec-14}
 
   269 
   283 
   270   Find out the eigen values and eigen vectors of the matrix \texttt{m5}.
 
   284   Find out the eigen values and eigen vectors of the matrix \texttt{m5}.
 
   271 \begin{verbatim}
 
   285 \begin{verbatim}
 
   272    
   286    
   273 \end{verbatim}
 
   287 \end{verbatim}
 
   298 
   312 
   299 \end{itemize}
 
   313 \end{itemize}
 
   300 \end{frame}
 
   314 \end{frame}
 
   301 \begin{frame}[fragile]
 
   315 \begin{frame}[fragile]
 
   302 \frametitle{Singular Value Decomposition (\texttt{svd})}
 
   316 \frametitle{Singular Value Decomposition (\texttt{svd})}
 
   303 \label{sec-13}
 
   317 \label{sec-15}
 
   304 
   318 
   305     $M = U \Sigma V^*$
 
   319     $M = U \Sigma V^*$
 
   306 \begin{itemize}
 
   320 \begin{itemize}
 
   307 \item U, an \texttt{mXm} unitary matrix over K.
 
   321 \item U, an \texttt{mXm} unitary matrix over K.
 
   308 \item $\Sigma$
 
   322 \item $\Sigma$
 
   316      In []: svd(m5)
 
   330      In []: svd(m5)
 
   317 \end{verbatim}
 
   331 \end{verbatim}
 
   318 \end{frame}
 
   332 \end{frame}
 
   319 \begin{frame}
 
   333 \begin{frame}
 
   320 \frametitle{Summary}
 
   334 \frametitle{Summary}
 
   321 \label{sec-14}
 
   335 \label{sec-16}
 
   322 
   336 
   323 \begin{itemize}
 
   337 \begin{itemize}
 
   324 \item Matrices
 
   338 \item Matrices
 
   325 
   339 
   326 \begin{itemize}
 
   340 \begin{itemize}
 
   335 \item Singular Value Decomposition (\texttt{svd()})
 
   349 \item Singular Value Decomposition (\texttt{svd()})
 
   336 \end{itemize}
 
   350 \end{itemize}
 
   337 \end{frame}
 
   351 \end{frame}
 
   338 \begin{frame}
 
   352 \begin{frame}
 
   339 \frametitle{Thank you!}
 
   353 \frametitle{Thank you!}
 
   340 \label{sec-15}
 
   354 \label{sec-17}
 
   341 
   355 
   342   \begin{block}{}
 
   356   \begin{block}{}
 
   343   \begin{center}
 
   357   \begin{center}
 
   344   This spoken tutorial has been produced by the
 
   358   This spoken tutorial has been produced by the
 
   345   \textcolor{blue}{FOSSEE} team, which is funded by the 
   359   \textcolor{blue}{FOSSEE} team, which is funded by the
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