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    21 \usepackage{listings}

    22 \lstset{language=Python, basicstyle=\ttfamily\bfseries,

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    25 \providecommand{\alert}[1]{\textbf{#1}}

    26 

    27 \title{Matrices}

    28 \author{FOSSEE}

    29 \date{}

    30 

    31 \usetheme{Warsaw}\usecolortheme{default}\useoutertheme{infolines}\setbeamercovered{transparent}

    32 \begin{document}

    33 

    34 \maketitle

    35 

    36 

    37 

    38 

    39 

    40 

    41 

    42 

    43 

    44 \begin{frame}

    45 \frametitle{Outline}

    46 \label{sec-1}

    47 

    48 \begin{itemize}

    49 \item Creating Matrices

    50 

    51 \begin{itemize}

    52 \item using direct data

    53 \item converting a list

    54 \end{itemize}

    55 

    56 \item Matrix operations

    57 \item Inverse of matrix

    58 \item Determinant of matrix

    59 \item Eigen values and Eigen vectors of matrices

    60 \item Norm of matrix

    61 \item Singular Value Decomposition of matrices

    62 \end{itemize}

    63 \end{frame}

    64 \begin{frame}[fragile]

    65 \frametitle{Creating a matrix}

    66 \label{sec-2}

    67 

    68 \begin{itemize}

    69 \item Creating a matrix using direct data

    70 \end{itemize}

    71 

    72 \begin{verbatim}

    73    In []: m1 = matrix([1, 2, 3, 4])

    74 \end{verbatim}

    75 

    76 \begin{itemize}

    77 \item Creating a matrix using lists

    78 \end{itemize}

    79 

    80 \begin{verbatim}

    81    In []: l1 = [[1,2,3,4],[5,6,7,8]]

    82    In []: m2 = matrix(l1)

    83 \end{verbatim}

    84 

    85 \begin{itemize}

    86 \item A matrix is basically an array

    87 \end{itemize}

    88 

    89 \begin{verbatim}

    90    In []: m3 = array([[5,6,7,8],[9,10,11,12]])

    91 \end{verbatim}

    92 \end{frame}

    93 \begin{frame}[fragile]

    94 \frametitle{Matrix operations}

    95 \label{sec-3}

    96 

    97 \begin{itemize}

    98 \item Element-wise addition (both matrix should be of order \texttt{mXn})

    99 \begin{verbatim}

   100      In []: m3 + m2

   101 \end{verbatim}

   102 

   103 \item Element-wise subtraction (both matrix should be of order \texttt{mXn})

   104 \begin{verbatim}

   105      In []: m3 - m2

   106 \end{verbatim}

   107 

   108 \end{itemize}

   109 \end{frame}

   110 \begin{frame}[fragile]

   111 \frametitle{Matrix Multiplication}

   112 \label{sec-4}

   113 

   114 \begin{itemize}

   115 \item Matrix Multiplication

   116 \begin{verbatim}

   117      In []: m3 * m2

   118      Out []: ValueError: objects are not aligned

   119 \end{verbatim}

   120 

   121 \item Element-wise multiplication using \texttt{multiply()}

   122 \begin{verbatim}

   123      multiply(m3, m2)

   124 \end{verbatim}

   125 

   126 \end{itemize}

   127 \end{frame}

   128 \begin{frame}[fragile]

   129 \frametitle{Matrix Multiplication (cont'd)}

   130 \label{sec-5}

   131 

   132 \begin{itemize}

   133 \item Create two compatible matrices of order \texttt{nXm} and \texttt{mXr}

   134 \begin{verbatim}

   135      In []: m1.shape

   136 \end{verbatim}

   137 

   138 

   139 \begin{itemize}

   140 \item matrix m1 is of order \texttt{1 X 4}

   141 \end{itemize}

   142 

   143 \item Creating another matrix of order \texttt{4 X 2}

   144 \begin{verbatim}

   145      In []: m4 = matrix([[1,2],[3,4],[5,6],[7,8]])

   146 \end{verbatim}

   147 

   148 \item Matrix multiplication

   149 \begin{verbatim}

   150      In []: m1 * m4

   151 \end{verbatim}

   152 

   153 \end{itemize}

   154 \end{frame}

   155 \begin{frame}

   156 \frametitle{Recall from \texttt{array}}

   157 \label{sec-6}

   158 

   159 \begin{itemize}

   160 \item The functions

   161 

   162 \begin{itemize}

   163 \item \texttt{identity(n)} - 

   164       creates an identity matrix of order \texttt{nXn}

   165 \item \texttt{zeros((m,n))} - 

   166       creates a matrix of order \texttt{mXn} with 0's

   167 \item \texttt{zeros\_like(A)} - 

   168       creates a matrix with 0's similar to the shape of matrix \texttt{A}

   169 \item \texttt{ones((m,n))}

   170       creates a matrix of order \texttt{mXn} with 1's

   171 \item \texttt{ones\_like(A)}

   172       creates a matrix with 1's similar to the shape of matrix \texttt{A}

   173 \end{itemize}

   174 

   175 \end{itemize}

   176 

   177   Can also be used with matrices

   178 \end{frame}

   179 \begin{frame}[fragile]

   180 \frametitle{More matrix operations}

   181 \label{sec-7}

   182 

   183   Transpose of a matrix

   184 \begin{verbatim}

   185    In []: m4.T

   186 \end{verbatim}

   187 \end{frame}

   188 \begin{frame}[fragile]

   189 \frametitle{Exercise 1 : Frobenius norm \& inverse}

   190 \label{sec-8}

   191 

   192   Find out the Frobenius norm of inverse of a \texttt{4 X 4} matrix.

   193 \begin{verbatim}

   194    

   195 \end{verbatim}

   196 

   197   The matrix is

   198 \begin{verbatim}

   199    m5 = matrix(arange(1,17).reshape(4,4))

   200 \end{verbatim}

   201 

   202 \begin{itemize}

   203 \item Inverse of A,

   204 

   205 \begin{itemize}

   206 \item $A^{-1} = inv(A)$

   207 \end{itemize}

   208 

   209 \item Frobenius norm is defined as,

   210 

   211 \begin{itemize}

   212 \item $||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}$

   213 \end{itemize}

   214 

   215 \end{itemize}

   216 \end{frame}

   217 \begin{frame}[fragile]

   218 \frametitle{Exercise 2: Infinity norm}

   219 \label{sec-9}

   220 

   221   Find the infinity norm of the matrix \texttt{im5}

   222 \begin{verbatim}

   223    

   224 \end{verbatim}

   225 

   226 \begin{itemize}

   227 \item Infinity norm is defined as,

   228        $max([\sum_{i} abs(a_{i})^2])$

   229 \end{itemize}

   230 \end{frame}

   231 \begin{frame}[fragile]

   232 \frametitle{\texttt{norm()} method}

   233 \label{sec-10}

   234 

   235 \begin{itemize}

   236 \item Frobenius norm

   237 \begin{verbatim}

   238      In []: norm(im5)

   239 \end{verbatim}

   240 

   241 \item Infinity norm

   242 \begin{verbatim}

   243      In []: norm(im5, ord=inf)

   244 \end{verbatim}

   245 

   246 \end{itemize}

   247 \end{frame}

   248 \begin{frame}[fragile]

   249 \frametitle{Determinant}

   250 \label{sec-11}

   251 

   252   Find out the determinant of the matrix m5

   253 \begin{verbatim}

   254    

   255 \end{verbatim}

   256 

   257 \begin{itemize}

   258 \item determinant can be found out using

   259 

   260 \begin{itemize}

   261 \item \texttt{det(A)} - returns the determinant of matrix \texttt{A}

   262 \end{itemize}

   263 

   264 \end{itemize}

   265 \end{frame}

   266 \begin{frame}[fragile]

   267 \frametitle{eigen values \& eigen vectors}

   268 \label{sec-12}

   269 

   270   Find out the eigen values and eigen vectors of the matrix \texttt{m5}.

   271 \begin{verbatim}

   272    

   273 \end{verbatim}

   274 

   275 \begin{itemize}

   276 \item eigen values and vectors can be found out using

   277 \begin{verbatim}

   278      In []: eig(m5)

   279 \end{verbatim}

   280 

   281     returns a tuple of \emph{eigen values} and \emph{eigen vectors}

   282 \item \emph{eigen values} in tuple

   283 

   284 \begin{itemize}

   285 \item \texttt{In []: eig(m5)[0]}

   286 \end{itemize}

   287 

   288 \item \emph{eigen vectors} in tuple

   289 

   290 \begin{itemize}

   291 \item \texttt{In []: eig(m5)[1]}

   292 \end{itemize}

   293 

   294 \item Computing \emph{eigen values} using \texttt{eigvals()}

   295 \begin{verbatim}

   296      In []: eigvals(m5)

   297 \end{verbatim}

   298 

   299 \end{itemize}

   300 \end{frame}

   301 \begin{frame}[fragile]

   302 \frametitle{Singular Value Decomposition (\texttt{svd})}

   303 \label{sec-13}

   304 

   305     $M = U \Sigma V^*$

   306 \begin{itemize}

   307 \item U, an \texttt{mXm} unitary matrix over K.

   308 \item $\Sigma$

   309         , an \texttt{mXn} diagonal matrix with non-negative real numbers on diagonal.

   310 \item $V^*$

   311         , an \texttt{nXn} unitary matrix over K, denotes the conjugate transpose of V.

   312 \item SVD of matrix \texttt{m5} can be found out as,

   313 \end{itemize}

   314 

   315 \begin{verbatim}

   316      In []: svd(m5)

   317 \end{verbatim}

   318 \end{frame}

   319 \begin{frame}

   320 \frametitle{Summary}

   321 \label{sec-14}

   322 

   323 \begin{itemize}

   324 \item Matrices

   325 

   326 \begin{itemize}

   327 \item creating matrices

   328 \end{itemize}

   329 

   330 \item Matrix operations

   331 \item Inverse (\texttt{inv()})

   332 \item Determinant (\texttt{det()})

   333 \item Norm (\texttt{norm()})

   334 \item Eigen values \& vectors (\texttt{eig(), eigvals()})

   335 \item Singular Value Decomposition (\texttt{svd()})

   336 \end{itemize}

   337 \end{frame}

   338 \begin{frame}

   339 \frametitle{Thank you!}

   340 \label{sec-15}

   341 

   342   \begin{block}{}

   343   \begin{center}

   344   This spoken tutorial has been produced by the

   345   \textcolor{blue}{FOSSEE} team, which is funded by the 

   346   \end{center}

   347   \begin{center}

   348     \textcolor{blue}{National Mission on Education through \\

   349       Information \& Communication Technology \\ 

   350       MHRD, Govt. of India}.

   351   \end{center}  

   352   \end{block}

   353 

   354   

   355 \end{frame}

   356 

   357 \end{document}