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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[Matrices \& Equations]{Python for Science and Engg: 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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%% the beginning of each subsection:
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    \frametitle{Outline}
    \tableofcontents[currentsection,currentsubsection]
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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,  1,  2, -1],
                   [ 2,  5, -1, -9],
                   [ 2,  1, -1,  3],
                   [ 1, -3,  2,  7]])
In []: A
Out[]: 
matrix([[ 1,  1,  2, -1],
        [ 2,  5, -1, -9],
        [ 2,  1, -1,  3],
        [ 1, -3,  2,  7]])
\end{lstlisting}
\end{frame}

\subsection{Basic Operations}

\begin{frame}[fragile]
\frametitle{Transpose of a Matrix}
\begin{lstlisting}
In []: linalg.transpose(A)
Out[]:
matrix([[ 1,  2,  2,  1],
        [ 1,  5,  1, -3],
        [ 2, -1, -1,  2],
        [-1, -9,  3,  7]])
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
  \frametitle{Sum of all elements}
  \begin{lstlisting}
In []: linalg.sum(A)
Out[]: 12
  \end{lstlisting}
\end{frame}

\begin{frame}[fragile]
  \frametitle{Matrix Addition}
  \begin{lstlisting}
In []: B = matrix([[3,2,-1,5],
                   [2,-2,4,9],
                   [-1,0.5,-1,-7],
                   [9,-5,7,3]])
In []: linalg.add(A,B)
Out[]: 
matrix([[  4. ,   3. ,   1. ,   4. ],
        [  4. ,   3. ,   3. ,   0. ],
        [  1. ,   1.5,  -2. ,  -4. ],
        [ 10. ,  -8. ,   9. ,  10. ]])
  \end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Matrix Multiplication}
\begin{lstlisting}
In []: linalg.multiply(A, B)
Out[]: 
matrix([[  3. ,   2. ,  -2. ,  -5. ],
        [  4. , -10. ,  -4. , -81. ],
        [ -2. ,   0.5,   1. , -21. ],
        [  9. ,  15. ,  14. ,  21. ]])
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Inverse of a Matrix}
\begin{small}
\begin{lstlisting}
In []: linalg.inv(A)
Out[]: 
matrix([[-0.5 ,  0.55, -0.15,  0.7 ],
        [ 0.75, -0.5 ,  0.5 , -0.75],
        [ 0.5 , -0.15, -0.05, -0.1 ],
        [ 0.25, -0.25,  0.25, -0.25]])
\end{lstlisting}
\end{small}
\end{frame}

\begin{frame}[fragile]
\frametitle{Determinant}
\begin{lstlisting}
In []: det(A)
Out[66]: 80.0
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Eigen Values and Eigen Matrix}
\begin{small}
\begin{lstlisting}
In []: E = matrix([[3,2,4],[2,0,2],[4,2,3]])

In []: linalg.eig(E)
Out[]: 
(array([-1.,  8., -1.]),
 matrix([[-0.74535599,  0.66666667, -0.1931126 ],
        [ 0.2981424 ,  0.33333333, -0.78664085],
        [ 0.59628479,  0.66666667,  0.58643303]]))

In []: linalg.eigvals(E)
Out[]: array([-1.,  8., -1.])
\end{lstlisting}
\end{small}
\end{frame}

\begin{frame}[fragile]
\frametitle{Computing Norms}
\begin{lstlisting}
In []: linalg.norm(E)
Out[]: 8.1240384046359608
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
  \frametitle{Single Value Decomposition}
  \begin{small}
  \begin{lstlisting}
In [76]: linalg.svd(E)
Out[76]: 
(matrix(
[[ -6.66666667e-01,  -1.23702565e-16,   7.45355992e-01],
 [ -3.33333333e-01,  -8.94427191e-01,  -2.98142397e-01],
 [ -6.66666667e-01,   4.47213595e-01,  -5.96284794e-01]]),
 array([ 8.,  1.,  1.]),
 matrix([[-0.66666667, -0.33333333, -0.66666667],
        [-0.        ,  0.89442719, -0.4472136 ],
        [-0.74535599,  0.2981424 ,  0.59628479]]))
  \end{lstlisting}
  \end{small}
\inctime{15}
\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)
  \end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Solution:}
\begin{lstlisting}
In []: x
Out[]: 
array([[ 1.],
       [-2.],
       [-2.]])
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Let's check!}
\begin{lstlisting}
In []: Ax
Out[]: 
matrix([[  1.00000000e+00],
        [ -2.00000000e+00],
        [  2.22044605e-16]])
\end{lstlisting}
\begin{block}{}
The last term in the matrix is actually \alert{0}!\\
We can use \kwrd{allclose()} to check.
\end{block}
\begin{lstlisting}
In []: allclose(Ax, b)
Out[]: True
\end{lstlisting}
\inctime{15}
\end{frame}

\subsection{Exercises}

\begin{frame}[fragile]
\frametitle{Problem Set 4: Problem 4.1}
Solve the set of equations:
\begin{align*}
  x + y + 2z -w & = 3\\
  2x + 5y - z - 9w & = -3\\
  2x + y -z + 3w & = -11 \\
  x - 3y + 2z + 7w & = -5\\
\end{align*}
\inctime{10}
\end{frame}

\begin{frame}[fragile]
\frametitle{Solution}
Solution:
\begin{lstlisting}
\begin{align*}
  x & = -5\\
  y & = 2\\
  z & = 3\\
  w & = 0\\
\end{align*}
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Problem 4.2}

\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}