% Created 2010-10-12 Tue 14:28

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\title{Matrices}

\author{FOSSEE}

\date{}

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\begin{document}

\maketitle

\begin{frame}

\frametitle{Outline}

\label{sec-1}

\begin{itemize}

\item Creating Matrices

\begin{itemize}

\item using direct data

\item converting a list

\end{itemize}

\item Matrix operations

\item Inverse of matrix

\item Determinant of matrix

\item Eigen values and Eigen vectors of matrices

\item Norm of matrix

\item Singular Value Decomposition of matrices

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Creating a matrix}

\label{sec-2}

\begin{itemize}

\item Creating a matrix using direct data

\end{itemize}

\begin{verbatim}

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

\end{verbatim}

\begin{itemize}

\item Creating a matrix using lists

\end{itemize}

\begin{verbatim}

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

   In []: m2 = matrix(l1)

\end{verbatim}

\begin{itemize}

\item A matrix is basically an array

\end{itemize}

\begin{verbatim}

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

\end{verbatim}

\end{frame}

\begin{frame}[fragile]

\frametitle{Matrix operations}

\label{sec-3}

\begin{itemize}

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

\begin{verbatim}

     In []: m3 + m2

\end{verbatim}

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

\begin{verbatim}

     In []: m3 - m2

\end{verbatim}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Matrix Multiplication}

\label{sec-4}

\begin{itemize}

\item Matrix Multiplication

\begin{verbatim}

     In []: m3 * m2

     Out []: ValueError: objects are not aligned

\end{verbatim}

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

\begin{verbatim}

     multiply(m3, m2)

\end{verbatim}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Matrix Multiplication (cont'd)}

\label{sec-5}

\begin{itemize}

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

\begin{verbatim}

     In []: m1.shape

\end{verbatim}

\begin{itemize}

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

\end{itemize}

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

\begin{verbatim}

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

\end{verbatim}

\item Matrix multiplication

\begin{verbatim}

     In []: m1 * m4

\end{verbatim}

\end{itemize}

\end{frame}

\begin{frame}

\frametitle{Recall from \texttt{array}}

\label{sec-6}

\begin{itemize}

\item The functions

\begin{itemize}

\item \texttt{identity(n)} - 

      creates an identity matrix of order \texttt{nXn}

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

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

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

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

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

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

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

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

\end{itemize}

\end{itemize}

  Can also be used with matrices

\end{frame}

\begin{frame}[fragile]

\frametitle{More matrix operations}

\label{sec-7}

  Transpose of a matrix

\begin{verbatim}

   In []: m4.T

\end{verbatim}

\end{frame}

\begin{frame}[fragile]

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

\label{sec-8}

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

\begin{verbatim}

\end{verbatim}

  The matrix is

\begin{verbatim}

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

\end{verbatim}

\begin{itemize}

\item Inverse of A,

\begin{itemize}

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

\end{itemize}

\item Frobenius norm is defined as,

\begin{itemize}

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

\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Exercise 2: Infinity norm}

\label{sec-9}

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

\begin{verbatim}

\end{verbatim}

\begin{itemize}

\item Infinity norm is defined as,

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

\end{itemize}

\end{frame}

\begin{frame}[fragile]

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

\label{sec-10}

\begin{itemize}

\item Frobenius norm

\begin{verbatim}

     In []: norm(im5)

\end{verbatim}

\item Infinity norm

\begin{verbatim}

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

\end{verbatim}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Determinant}

\label{sec-11}

  Find out the determinant of the matrix m5

\begin{verbatim}

\end{verbatim}

\begin{itemize}

\item determinant can be found out using

\begin{itemize}

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

\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{eigen values \& eigen vectors}

\label{sec-12}

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

\begin{verbatim}

\end{verbatim}

\begin{itemize}

\item eigen values and vectors can be found out using

\begin{verbatim}

     In []: eig(m5)

\end{verbatim}

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

\item \emph{eigen values} in tuple

\begin{itemize}

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

\end{itemize}

\item \emph{eigen vectors} in tuple

\begin{itemize}

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

\end{itemize}

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

\begin{verbatim}

     In []: eigvals(m5)

\end{verbatim}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

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

\label{sec-13}

    $M = U \Sigma V^*$

\begin{itemize}

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

\item $\Sigma$

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

\item $V^*$

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

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

\end{itemize}

\begin{verbatim}

     In []: svd(m5)

\end{verbatim}

\end{frame}

\begin{frame}

\frametitle{Summary}

\label{sec-14}

\begin{itemize}

\item Matrices

\begin{itemize}

\item creating matrices

\end{itemize}

\item Matrix operations

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

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

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

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

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

\end{itemize}

\end{frame}

\begin{frame}

\frametitle{Thank you!}

\label{sec-15}

  \begin{block}{}

  \begin{center}

  This spoken tutorial has been produced by the

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

  \end{center}

  \begin{center}

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

      Information \& Communication Technology \\ 

      MHRD, Govt. of India}.

  \end{center}  

  \end{block}

\end{frame}

\end{document}