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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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\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}