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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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    \frametitle{Outline}
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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 = matrix([[1,2,3],
                     [4,5,6],
                     [7,8,9]])

  In []: a
  Out[]: 
  matrix([[1, 2, 3],
         [4, 5, 6],
         [7, 8, 9]])
\end{lstlisting}
\end{frame}

\subsection{Basic Operations}
\begin{frame}[fragile]
\frametitle{Inverse of a Matrix}

\begin{small}
\begin{lstlisting}
In []: linalg.inv(A)
Out[]: 
matrix([[ 0.07734807,  0.01657459,  0.32044199],
        [ 0.09944751, -0.12154696, -0.01657459],
        [-0.02762431, -0.07734807,  0.17127072]])

\end{lstlisting}
\end{small}
\end{frame}

\begin{frame}[fragile]
\frametitle{Determinant}
\begin{lstlisting}
  In []: linalg.det(a)
  Out[]: -9.5171266700777579e-16
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Computing Norms}
\begin{lstlisting}
  In []: linalg.norm(a)
  Out[]: 16.881943016134134
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Eigen Values and Eigen Matrix}
\begin{small}
\begin{lstlisting}
  In []: linalg.eigvals(a)
  Out[]: array([1.61168440e+01, -1.11684397e+00, -1.22196337e-15])

  In []: linalg.eig(a)
  Out[]: 
  (array([ 1.61168440e+01, -1.11684397e+00, -1.22196337e-15]),
   matrix([[-0.23197069, -0.78583024,  0.40824829],
          [-0.52532209, -0.08675134, -0.81649658],
          [-0.8186735 ,  0.61232756,  0.40824829]]))
\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
  \item Solving linear equations
  \end{itemize}
\end{frame}

\end{document}