15. Eigenvectors and Eigenvalues.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       "\n",
       "div.cell { /* Tunes the space between cells */\n",
       "margin-top:1em;\n",
       "margin-bottom:1em;\n",
       "}\n",
       "\n",
       "div.text_cell_render h1 { /* Main titles bigger, centered */\n",
       "font-size: 2.2em;\n",
       "line-height:1.4em;\n",
       "text-align:center;\n",
       "}\n",
       "\n",
       "div.text_cell_render h2 { /*  Parts names nearer from text */\n",
       "margin-bottom: -0.4em;\n",
       "}\n",
       "\n",
       "\n",
       "div.text_cell_render { /* Customize text cells */\n",
       "font-family: 'Arial';\n",
       "font-size:1.5em;\n",
       "line-height:1.4em;\n",
       "padding-left:3em;\n",
       "padding-right:3em;\n",
       "}\n",
       "\n",
       ".MathJax {\n",
       "    font-size: 70%;\n",
       "}\n",
       "</style>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import HTML\n",
    "css_file = './custom.css'\n",
    "HTML(open(css_file, \"r\").read())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Eigenvectors and Eigenvalues\n",
    "\n",
    "© 2018 Daniel Voigt Godoy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from intuitiveml.algebra.LinearAlgebra import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Definitions\n",
    "\n",
    "We should start with a ***recap*** of ***linear algebra***.\n",
    "\n",
    "### 1.0 Standard Basis\n",
    "\n",
    "The standard basis is composed of N orthogonal unit vectors (as the number of dimensions).\n",
    "\n",
    "In 3D:\n",
    "\n",
    "$$\n",
    "\\hat{i} = \n",
    " \\begin{bmatrix}\n",
    "   1 \\\\\n",
    "   0 \\\\\n",
    "   0\n",
    " \\end{bmatrix}\n",
    "\\\n",
    "\\hat{j} = \n",
    " \\begin{bmatrix}\n",
    "   0 \\\\\n",
    "   1 \\\\\n",
    "   0\n",
    " \\end{bmatrix}\n",
    "\\\n",
    "\\hat{k} = \n",
    " \\begin{bmatrix}\n",
    "   0 \\\\\n",
    "   0 \\\\\n",
    "   1\n",
    " \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fd/3D_Vector.svg/254px-3D_Vector.svg.png)\n",
    "<center>Source: Wikipedia</center>\n",
    "\n",
    "But we'll stick with 2D vectors for the remainder of this lesson, so we'll be using:\n",
    "\n",
    "$$\n",
    "\\hat{i} = \n",
    " \\begin{bmatrix}\n",
    "   1 \\\\\n",
    "   0\n",
    " \\end{bmatrix}\n",
    "\\\n",
    "\\hat{j} = \n",
    " \\begin{bmatrix}\n",
    "   0 \\\\\n",
    "   1\n",
    " \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    "### 1.1 Matrix Vector Multiplication\n",
    "\n",
    "For a 2D data point $(x_1, x_2) \\\\ $ represented as a vector $\\boldsymbol{v}$, we can multiply it by a matrix $\\boldsymbol{A}$ as follows:\n",
    "\n",
    "$$\n",
    "\\boldsymbol{Av} = \n",
    "\\begin{bmatrix}\n",
    "   a_{11} & a_{12} \\\\\n",
    "   a_{21} & a_{22}\n",
    " \\end{bmatrix}\n",
    " \\begin{bmatrix}\n",
    "   x_1 \\\\\n",
    "   x_2\n",
    " \\end{bmatrix}\n",
    " =\n",
    " x_1\n",
    " \\begin{bmatrix}\n",
    "   a_{11} \\\\\n",
    "   a_{21}\n",
    " \\end{bmatrix}\n",
    " +\n",
    " x_2\n",
    " \\begin{bmatrix}\n",
    "   a_{12} \\\\\n",
    "   a_{22}\n",
    " \\end{bmatrix} \n",
    "$$\n",
    "\n",
    "Notice that this representation is ***different*** from what is usually learned in school!\n",
    "\n",
    "We are taking the ***first coordinate*** of our data point $x_1$ and multiplying by the ***first column*** of our matrix. Then we do the same for the second coordinate and column and ***add them up***.\n",
    "\n",
    "What if we perform the multiplication on the ***standard basis vectors*** $\\hat{i}$ and $\\hat{j}$?\n",
    "\n",
    "$$\n",
    "\\boldsymbol{A\\hat{i}} = \n",
    "\\begin{bmatrix}\n",
    "   a_{11} & a_{12} \\\\\n",
    "   a_{21} & a_{22}\n",
    " \\end{bmatrix}\n",
    " \\begin{bmatrix}\n",
    "   1 \\\\\n",
    "   0\n",
    " \\end{bmatrix}\n",
    " =\n",
    " 1\n",
    " \\begin{bmatrix}\n",
    "   a_{11} \\\\\n",
    "   a_{21}\n",
    " \\end{bmatrix}\n",
    " +\n",
    " 0\n",
    " \\begin{bmatrix}\n",
    "   a_{12} \\\\\n",
    "   a_{22}\n",
    " \\end{bmatrix} \n",
    " =\n",
    "  \\begin{bmatrix}\n",
    "   a_{11} \\\\\n",
    "   a_{21}\n",
    " \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\boldsymbol{A\\hat{j}} = \n",
    "\\begin{bmatrix}\n",
    "   a_{11} & a_{12} \\\\\n",
    "   a_{21} & a_{22}\n",
    " \\end{bmatrix}\n",
    " \\begin{bmatrix}\n",
    "   0 \\\\\n",
    "   1\n",
    " \\end{bmatrix}\n",
    " =\n",
    " 0\n",
    " \\begin{bmatrix}\n",
    "   a_{11} \\\\\n",
    "   a_{21}\n",
    " \\end{bmatrix}\n",
    " +\n",
    " 1\n",
    " \\begin{bmatrix}\n",
    "   a_{12} \\\\\n",
    "   a_{22}\n",
    " \\end{bmatrix} \n",
    " =\n",
    "  \\begin{bmatrix}\n",
    "   a_{12} \\\\\n",
    "   a_{22}\n",
    " \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "The ***new*** $\\hat{i}$ equals the ***first column***.\n",
    "\n",
    "The ***new*** $\\hat{j}$ equals the ***second column***.\n",
    "\n",
    "So, we can think of our matrix $\\boldsymbol{A}$ as a representation of the ***transformed basis vectors***.\n",
    "\n",
    "### 1.2 Exercise\n",
    "\n",
    "Time to try it yourself!\n",
    "\n",
    "There is a standard ***grid***. You can perform ***linear transformations*** on it by defining a ***transformation matrix***\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "   w_{11} & w_{12} \\\\\n",
    "   w_{21} & w_{22}\n",
    " \\end{bmatrix}\n",
    " $$ using the corresponding sliders to each one of its entries.\n",
    " \n",
    "It shows the ***basis vectors*** $\\hat{i}$ and $\\hat{j}$ both ***before (black)*** and ***after (red)*** the transformation.\n",
    " \n",
    "The ***step*** slider allows you to ***gradually*** perform the transformation: \n",
    " - at ***zero***, the grid is on its ***initial state***\n",
    " - at ***20***, the grid is ***fully transformed***.\n",
    " \n",
    "Use the sliders to play with different configurations. Try performing a ***rotation***, for instance: you need to figure out where the ***transformed basis vectors SHOULD land***, and set the ***transformation matrix*** accordingly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "transf = np.array([[1, 0], [0, 1]])\n",
    "eo2 = plotEigen(transf)\n",
    "vb2 = VBox(build_figure(eo2, basis=True), layout={'align_items': 'center'})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "816ec31ff39d46d2b24a5d1910cb4ed2",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "VBox(children=(FigureWidget({\n",
       "    'data': [{'marker': {'color': 'red', 'symbol': 'triangle-up'},\n",
       "             …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vb2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3 Linear Transformations\n",
    "\n",
    "![](https://imgs.xkcd.com/comics/matrix_transform.png)\n",
    "<center>Source: <a href=\"https://xkcd.com/184/\">XKCD</a></center>\n",
    "\n",
    "Some matrices represent what is called a ***linear transformation***, as long as they do preserve some of the characteristics of the original grid:\n",
    "- all lines remain lines (even ***diagonals***)\n",
    "- origin remains fixed in place\n",
    "- grid lines remain ***parallel*** and ***evenly spaced***\n",
    "\n",
    "So, no ***squishing*** of any kind!\n",
    "\n",
    "Some examples of linear transformations:\n",
    "\n",
    "### Rotation\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "   0 & -1 \\\\\n",
    "   1 & 0\n",
    " \\end{bmatrix}\n",
    " \\begin{bmatrix}\n",
    "   x_1 \\\\\n",
    "   x_2\n",
    " \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "### Shear\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "   1 & 1 \\\\\n",
    "   0 & 1\n",
    " \\end{bmatrix}\n",
    " \\begin{bmatrix}\n",
    "   x_1 \\\\\n",
    "   x_2\n",
    " \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "### Composition (First rotation, then shear)\n",
    "\n",
    "What if we want to first rotate and then shear? We can perform a sequence of matrix multiplications, we just need to be careful to do it in ***the right order***, that is, ***last operation to the left***.\n",
    "\n",
    "$$\n",
    "\\underbrace{\n",
    "\\begin{bmatrix}\n",
    "   1 & 1 \\\\\n",
    "   0 & 1\n",
    " \\end{bmatrix}\n",
    "}_{Shear}\n",
    "\\underbrace{\n",
    "\\begin{bmatrix}\n",
    "   0 & -1 \\\\\n",
    "   1 & 0\n",
    " \\end{bmatrix}\n",
    "}_{Rotation}\n",
    " \\begin{bmatrix}\n",
    "   x_1 \\\\\n",
    "   x_2\n",
    " \\end{bmatrix}\n",
    " =\n",
    " \\begin{bmatrix}\n",
    "   1 & -1 \\\\\n",
    "   1 & 0\n",
    " \\end{bmatrix}\n",
    " \\begin{bmatrix}\n",
    "   x_1 \\\\\n",
    "   x_2\n",
    " \\end{bmatrix}\n",
    " =\n",
    " x\n",
    " \\begin{bmatrix}\n",
    "   1 \\\\\n",
    "   1\n",
    " \\end{bmatrix}\n",
    " +\n",
    " y\n",
    " \\begin{bmatrix}\n",
    "   -1 \\\\\n",
    "   0\n",
    " \\end{bmatrix} \n",
    "$$\n",
    "\n",
    "### 1.4 Affine Transformations\n",
    "\n",
    "What if we want to ***shift the origin*** to a different coordinate, say (1, 1)?\n",
    "\n",
    "We could easily do that by ***adding a vector*** $ \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}$  after performing the ***linear transformation*** (matrix multiplication).\n",
    "\n",
    "But then the result is not a linear transformation anymore, it is going to be an ***affine transformation***.\n",
    "\n",
    "### 1.5 Nonsquare Matrix\n",
    "\n",
    "What if the ***transformation matrix*** is ***nonsquare***? Can it work? What is it good for?\n",
    "\n",
    "Yes, for ***projections***. If we have a ***3D data*** and we want to map it into a ***2D space***, we can use a ***2 x 3 transformation matrix***.\n",
    "\n",
    "Let's try multiplying such a matrix by a vector:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "   a_{11} & a_{12} & a_{13} \\\\\n",
    "   a_{21} & a_{22} & a_{23}\n",
    " \\end{bmatrix}\n",
    " \\begin{bmatrix}\n",
    "   x_1 \\\\\n",
    "   x_2 \\\\\n",
    "   x_3\n",
    " \\end{bmatrix}\n",
    " =\n",
    " \\\\\n",
    " x_1\n",
    " \\begin{bmatrix}\n",
    "   a_{11} \\\\\n",
    "   a_{21}\n",
    " \\end{bmatrix}\n",
    " +\n",
    " x_2\n",
    " \\begin{bmatrix}\n",
    "   a_{12} \\\\\n",
    "   a_{22}\n",
    " \\end{bmatrix} \n",
    " +\n",
    " x_3\n",
    " \\begin{bmatrix}\n",
    "   a_{13} \\\\\n",
    "   a_{23}\n",
    " \\end{bmatrix} \n",
    "$$\n",
    "\n",
    "It is easy to see that it has to have 3 columns because our data is 3D and each feature, $x_1$, $x_2$ and $x_3$ is going to multiply a vector. And since we want our result in 2D, it is only logical to add up 2D vectors, therefore, 2 rows.\n",
    "\n",
    "\n",
    "### 1.6 Eigenvectors and Eigenvalues\n",
    "\n",
    "If we perform a ***rotation***, ***all lines*** are rotating together, not a single one will stay with its ***original orientation***.\n",
    "\n",
    "But if we perform a ***shear***, what does happen to a vector on the ***x axis***? Does it ***move***? ***No***, and that's the reason why this vector is an ***eigenvector***.\n",
    "\n",
    "An ***eigenvector*** is a vector that remains on its own ***span*** after a ***linear transformation***.\n",
    "\n",
    "In other words, it is a ***vector*** that ***does not change its orientation***, except for maybe ***pointing to the opposite side*** and ***stretching / shrinking*** in size.\n",
    "\n",
    "The factor by which the vector ***stretches / shrinks*** is the ***eigenvalue*** $\\lambda$.\n",
    "\n",
    "Mathematically speaking, for a ***linear transformation*** $\\boldsymbol{A}$, an ***eigenvector*** $\\boldsymbol{v}$ \n",
    "and an ***eigenvalue*** $\\lambda$, the following holds:\n",
    "\n",
    "$$\n",
    "\\boldsymbol{Av} = \\lambda\\boldsymbol{v}\n",
    "\\\\\n",
    "\\boldsymbol{Av} = (\\lambda\\boldsymbol{I})\\boldsymbol{v}\n",
    "\\\\\n",
    "(\\boldsymbol{A}-\\lambda\\boldsymbol{I})\\boldsymbol{v} = 0\n",
    "\\\\\n",
    "|\\boldsymbol{A}-\\lambda\\boldsymbol{I}| = 0\n",
    "$$\n",
    "\n",
    "***IMPORTANT***: A ***transformation*** in 2D ***does not necessarily*** have two eigenvectors (disregarding complex eigenvectors, that is)!\n",
    "\n",
    "An ***easier*** way of thinking of ***eigenvectors*** is to consider them ***axes of rotation***. This idea should be more clear when you try the ***experiment***."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Experiment\n",
    "\n",
    "Time to try it yourself!\n",
    "\n",
    "There is a standard ***grid***. You can perform ***linear transformations*** on it by defining a ***transformation matrix***\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "   w_{11} & w_{12} \\\\\n",
    "   w_{21} & w_{22}\n",
    " \\end{bmatrix}\n",
    " $$\n",
    "using the corresponding sliders to each one of its entries.\n",
    "  \n",
    "It shows the ***eigevectors*** both ***before (black)*** and ***after (red)*** the transformation.\n",
    "\n",
    " The ***step*** slider allows you to ***gradually*** perform the transformation: \n",
    " - at ***zero***, the grid is on its ***initial state***\n",
    " - at ***20***, the grid is ***fully transformed***.\n",
    " \n",
    "Use the sliders to play with different configurations and answer the ***questions*** below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "transf = np.array([[1, 0], [0, 1]])\n",
    "eo = plotEigen(transf)\n",
    "vb = VBox(build_figure(eo), layout={'align_items': 'center'})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "934136e4120f43d2a8011d726d067e15",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "VBox(children=(FigureWidget({\n",
       "    'data': [{'marker': {'color': 'red', 'symbol': 'triangle-up'},\n",
       "             …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vb"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Questions\n",
    "\n",
    "1. Set ***step to zero*** and the weights to ***shear***: $$\\begin{bmatrix}\n",
    "   1 & 1 \\\\\n",
    "   0 & 1\n",
    " \\end{bmatrix}\n",
    "$$ \n",
    "    - what are the calculated ***eigenvalues***? \n",
    "    - what do you expect to happen to the ***size*** of the eigenvectors?\n",
    "    - now move the ***step to 20*** - what happened to the transformed eigenvectors? \n",
    "\n",
    "\n",
    "2. Set ***step to zero*** and the weights to: $$\\begin{bmatrix}\n",
    "   1 & 1 \\\\\n",
    "   0 & 0\n",
    " \\end{bmatrix}\n",
    " $$\n",
    "    - what are the calculated ***eigenvalues***? \n",
    "    - what do you expect to happen to the ***size*** of the eigenvectors?\n",
    "    - pay attention to the ***tip of eigenvector 2 (in black)*** - at which square does it land?\n",
    "    - now move the ***step to 5*** - where the ***tipo of transformed eigenvector 2 (in red)*** lands?\n",
    "    - keep moving the ***step until 20*** - what happened to the transformed eigenvectors?\n",
    "\n",
    "\n",
    "3. Set ***step to zero*** and the weights to ***rotation***: $$\\begin{bmatrix}\n",
    "   0 & -1 \\\\\n",
    "   1 & 0\n",
    " \\end{bmatrix}\n",
    " $$\n",
    "    - what are the calculated ***eigenvalues***? ***How*** are they different?\n",
    "    - now move the ***step to 20*** - what happened to the transformed eigenvectors?\n",
    "    - is this transformation ***different*** than the other two? Why?\n",
    "\n",
    "\n",
    "4. Set ***step to zero*** and the weights to: $$\\begin{bmatrix}\n",
    "   1 & -1 \\\\\n",
    "   -1 & 0\n",
    " \\end{bmatrix}\n",
    " $$\n",
    "    - what are the calculated ***eigenvalues***?\n",
    "    - can you tell ***how*** the ***eigenvectors*** are going to change from the ***eigenvalues alone***?\n",
    "    - now move the ***step to 20*** - what happened to the transformed eigenvectors?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Numpy\n",
    "\n",
    "[numpy.linalg.eig](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.linalg.eig.html)\n",
    "\n",
    "[numpy.linalg.svd](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.linalg.svd.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. More Resources\n",
    "\n",
    "[Essence of Linear Algebra Series](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)\n",
    "\n",
    "[Essence of Linear Algebra - Eigenvector and eigenvalues](https://www.youtube.com/watch?v=PFDu9oVAE-g&index=14&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&t=0s)\n",
    "\n",
    "[INTERACTIVE Immersive Linear Algebra](http://immersivemath.com/ila/index.html)\n",
    "\n",
    "[Linear algebra cheat sheet for deep learning](https://towardsdatascience.com/linear-algebra-cheat-sheet-for-deep-learning-cd67aba4526c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### This material is copyright Daniel Voigt Godoy and made available under the Creative Commons Attribution (CC-BY) license ([link](https://creativecommons.org/licenses/by/4.0/)). \n",
    "\n",
    "#### Code is also made available under the MIT License ([link](https://opensource.org/licenses/MIT))."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<script>\n",
       "  function code_toggle() {\n",
       "    if (code_shown){\n",
       "      $('div.input').hide('500');\n",
       "      $('#toggleButton').val('Show Code')\n",
       "    } else {\n",
       "      $('div.input').show('500');\n",
       "      $('#toggleButton').val('Hide Code')\n",
       "    }\n",
       "    code_shown = !code_shown\n",
       "  }\n",
       "\n",
       "  $( document ).ready(function(){\n",
       "    code_shown=false;\n",
       "    $('div.input').hide()\n",
       "  });\n",
       "</script>\n",
       "<form action=\"javascript:code_toggle()\"><input type=\"submit\" id=\"toggleButton\" value=\"Show Code\"></form>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import HTML\n",
    "HTML('''<script>\n",
    "  function code_toggle() {\n",
    "    if (code_shown){\n",
    "      $('div.input').hide('500');\n",
    "      $('#toggleButton').val('Show Code')\n",
    "    } else {\n",
    "      $('div.input').show('500');\n",
    "      $('#toggleButton').val('Hide Code')\n",
    "    }\n",
    "    code_shown = !code_shown\n",
    "  }\n",
    "\n",
    "  $( document ).ready(function(){\n",
    "    code_shown=false;\n",
    "    $('div.input').hide()\n",
    "  });\n",
    "</script>\n",
    "<form action=\"javascript:code_toggle()\"><input type=\"submit\" id=\"toggleButton\" value=\"Show Code\"></form>''')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}