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%Tutorial slides on Python.

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% Author: FOSSEE 

% Copyright (c) 2009, FOSSEE, IIT Bombay

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% Title page

\title[Basic Python]{Matrices, Solution of equations}

\author[FOSSEE] {FOSSEE}

\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}

\date[] {31, October 2009\\Day 1, Session 4}

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\begin{document}

\begin{frame}

  \titlepage

\end{frame}

\begin{frame}

  \frametitle{Outline}

  \tableofcontents

%  \pausesections

\end{frame}

\section{Matrices}

\begin{frame}

\frametitle{Matrices: Introduction}

We looked at the Van der Monde matrix in the previous session,\\ 

let us now look at matrices in a little more detail.

\end{frame}

\subsection{Initializing}

\begin{frame}[fragile]

\frametitle{Matrices: Initializing}

\begin{lstlisting}

In []: A = ([[5, 2, 4], 

            [-3, 6, 2], 

            [3, -3, 1]])

In []: A

Out[]: [[5, 2, 4], 

        [-3, 6, 2], 

        [3, -3, 1]]

\end{lstlisting}

\end{frame}

\subsection{Basic Operations}

\begin{frame}[fragile]

\frametitle{Transpose of a Matrix}

\begin{lstlisting}

In []: linalg.transpose(A)

Out[]: 

matrix([[ 5, -3,  3],

        [ 2,  6, -3],

        [ 4,  2,  1]])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Sum of all elements}

  \begin{lstlisting}

In []: linalg.sum(A)

Out[]: 17

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Matrix Addition}

  \begin{lstlisting}

In []: B = matrix([[3,2,-1],

                   [2,-2,4],

                   [-1, 0.5, -1]])

In []: linalg.add(A, B)

Out[]: 

matrix([[ 8. ,  4. ,  3. ],

        [-1. ,  4. ,  6. ],

        [ 2. , -2.5,  0. ]])

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Matrix Multiplication}

\begin{lstlisting}

In []: linalg.multiply(A, B)

Out[]: 

matrix([[ 15. ,   4. ,  -4. ],

        [ -6. , -12. ,   8. ],

        [ -3. ,  -1.5,  -1. ]])

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Inverse of a Matrix}

\begin{small}

\begin{lstlisting}

In []: linalg.inv(A)

Out[]: 

array([[ 0.28571429, -0.33333333, -0.47619048],

       [ 0.21428571, -0.16666667, -0.52380952],

       [-0.21428571,  0.5       ,  0.85714286]])

\end{lstlisting}

\end{small}

\end{frame}

\begin{frame}[fragile]

\frametitle{Determinant}

\begin{lstlisting}

In []: det(A)

Out[]: 42.0

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Eigen Values and Eigen Matrix}

\begin{small}

\begin{lstlisting}

In []: linalg.eig(A)

Out[]: 

(array([ 7.,  2.,  3.]),

 matrix([[-0.57735027,  0.42640143,  0.37139068],

        [ 0.57735027,  0.63960215,  0.74278135],

        [-0.57735027, -0.63960215, -0.55708601]]))

\end{lstlisting}

\end{small}

\end{frame}

\begin{frame}[fragile]

\frametitle{Computing Norms}

\begin{lstlisting}

  In []: linalg.norm(A)

  Out[]: 10.63014581273465

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Single Value Decomposition}

  \begin{small}

  \begin{lstlisting}

In []: linalg.svd(A)

Out[]: 

(matrix([[-0.13391246, -0.94558684, -0.29653495],

        [ 0.84641267, -0.26476432,  0.46204486],

        [-0.51541542, -0.18911737,  0.83581192]]),

 array([ 7.96445022,  7.        ,  0.75334767]),

 matrix([[-0.59703387,  0.79815896,  0.08057807],

        [-0.64299905, -0.41605821, -0.64299905],

        [-0.47969029, -0.43570384,  0.7616163 ]]))

  \end{lstlisting}

  \end{small}

\end{frame}

\section{Solving linear equations}

\begin{frame}[fragile]

\frametitle{Solution of equations}

Consider,

  \begin{align*}

    3x + 2y - z  & = 1 \\

    2x - 2y + 4z  & = -2 \\

    -x + \frac{1}{2}y -z & = 0

  \end{align*}

Solution:

  \begin{align*}

    x & = 1 \\

    y & = -2 \\

    z & = -2

  \end{align*}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving using Matrices}

Let us now look at how to solve this using \kwrd{matrices}

  \begin{lstlisting}

    In []: A = matrix([[3,2,-1],

                       [2,-2,4],

                       [-1, 0.5, -1]])

    In []: b = matrix([[1], [-2], [0]])

    In []: x = linalg.solve(A, b)

    In []: Ax = dot(A, x)

    In []: allclose(Ax, b)

    Out[]: True

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution:}

\begin{lstlisting}

In []: x

Out[]: 

array([[ 1.],

       [-2.],

       [-2.]])

In []: Ax

Out[]: 

matrix([[  1.00000000e+00],

        [ -2.00000000e+00],

        [  2.22044605e-16]])

\end{lstlisting}

\end{frame}

\section{Summary}

\begin{frame}

  \frametitle{Summary}

So what did we learn??

  \begin{itemize}

  \item Matrices

    \begin{itemize}

      \item Transpose

      \item Addition

      \item Multiplication

      \item Inverse of a matrix

      \item Determinant

      \item Eigen values and Eigen matrix

      \item Norms

      \item Single Value Decomposition

    \end{itemize}

  \item Solving linear equations

  \end{itemize}

\end{frame}

\end{document}