{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Rotation Matricies\n",
    "Detailed instructions to submit ipynb files can be found on Piazza Post @17.\n",
    "Please save your hw6.ipynb as a pdf and attach it to your written homework. This combined pdf will be submitted as hw6.pdf.\n",
    "Submit both hw6.pdf and hw6.ipynb through glookup."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.rcParams['figure.figsize'] = (30.0, 6.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fill in the function below so that it constructs a rotation matrix from an input angle, theta, and a scaling factor, r\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#Returns a rotation matrix for a desired rotation and scaling\n",
    "# theta is the degrees of rotation\n",
    "# r is the scaling factor of the rotation matrix\n",
    "def rotation_matrix(theta,r):\n",
    "   \n",
    "    # Represent the rotation matrix\n",
    "    r_matrix = #TODO\n",
    "  \n",
    "    return r_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eigenvalues of Rotation Matricies\n",
    "Use your matrix construction function to construct the specified matricies below. Fill in the #TODO lines with your code to do this. Note, it is best to represent the matrix as an np.array object since other code assumes the matricies will be in that form.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mat0 eigenvalues: [ 0.70710678+0.70710678j  0.70710678-0.70710678j]\n",
      "mat0 eigenvalues magnitudes: [ 1.  1.]\n",
      "mat1 eigenvalues: [ 0.88388348+0.88388348j  0.88388348-0.88388348j]\n",
      "mat1 eigenvalues magnitudes: [ 1.25  1.25]\n",
      "mat2 eigenvalues: [ 0.35355339+0.35355339j  0.35355339-0.35355339j]\n",
      "mat2 eigenvalues magnitudes: [ 0.5  0.5]\n"
     ]
    }
   ],
   "source": [
    "############################################\n",
    "#Construct a matrix for theta= pi/4, r = 1\n",
    "\n",
    "mat0=#TODO\n",
    "\n",
    "#calculate the eigenvalues and eigenvectors of mat0\n",
    "\n",
    "eig0=#TODO\n",
    "\n",
    "#Caculate the magnitude of the eigenvalues\n",
    "\n",
    "mag0=#TODO\n",
    "\n",
    "############################################\n",
    "#construct a matrix for theta = pi/4, r=1.25\n",
    "\n",
    "mat1=#TODO\n",
    "\n",
    "#calculate the eigenvalues and eigenvectors for mat1\n",
    "\n",
    "eig1=#TODO\n",
    "\n",
    "#Caculate the magnitude of the eigenvalues\n",
    "\n",
    "mag1=#TODO\n",
    "\n",
    "############################################\n",
    "#construct a matrix for theta = pi/4, r=0.5\n",
    "\n",
    "mat2=#TODO\n",
    "\n",
    "#calculate the eigenvalues and eigenvectors for mat2\n",
    "\n",
    "eig2= #TODO\n",
    "\n",
    "#Caculate the magnitude of the eigenvalues\n",
    "mag2=#TODO\n",
    "\n",
    "############################################\n",
    "\n",
    "\n",
    "print \"mat0 eigenvalues: \"+str(eig0)  \n",
    "print \"mat0 eigenvalues magnitudes: \"+str(mag0)\n",
    "print \"mat1 eigenvalues: \"+str(eig1)\n",
    "print \"mat1 eigenvalues magnitudes: \"+str(mag1)\n",
    "print \"mat2 eigenvalues: \"+str(eig2)\n",
    "print \"mat2 eigenvalues magnitudes: \"+str(mag2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rotation of Vectors Using Rotation Matricies \n",
    "Fill in the #TODO items to rotate the vector x1. Note, the plotting code assumes that the resulting vecotors are np.array objects"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109eaf6d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "x1=np.array([1,0])\n",
    "#Apply mat0 once to x1\n",
    "x1_r1=#TODO\n",
    "\n",
    "#Apply mat0 twice to x1\n",
    "x1_r2=#TODO\n",
    "\n",
    "#Apply mat0 three times to x1\n",
    "x1_r3=#TODO\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "#this code will plot your rotated state vectors\n",
    "fig =plt.figure()\n",
    "plt.subplot(141)\n",
    "plt.arrow(0,0,x1[0],x1[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('Original Vector')\n",
    "plt.subplot(142)\n",
    "plt.arrow(0,0,x1_r1[0],x1_r1[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('1 Rotation')\n",
    "plt.subplot(143)\n",
    "plt.arrow(0,0,x1_r2[0],x1_r2[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('2 Rotations')\n",
    "plt.subplot(144)\n",
    "plt.arrow(0,0,x1_r3[0],x1_r3[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('3 Rotations')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we will see what happens when we rotate this state vector by our second rotation matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109241050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1=np.array([1,0])\n",
    "#Apply mat1 once to x1\n",
    "x1_r1=#TODO\n",
    "\n",
    "#Apply mat1 twice to x1\n",
    "x1_r2=#TODO\n",
    "\n",
    "#Apply mat1 three times to x1\n",
    "x1_r3=#TODO\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "#this code will plot your rotated state vectors\n",
    "plt.close('all')\n",
    "plt.figure()\n",
    "plt.subplot(141)\n",
    "plt.arrow(0,0,x1[0],x1[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('Original Vector')\n",
    "plt.subplot(142)\n",
    "plt.arrow(0,0,x1_r1[0],x1_r1[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('1 Rotation')\n",
    "plt.subplot(143)\n",
    "plt.arrow(0,0,x1_r2[0],x1_r2[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('2 Rotations')\n",
    "plt.subplot(144)\n",
    "plt.arrow(0,0,x1_r3[0],x1_r3[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('3 Rotations')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's rotate x1 with mat2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109bcacd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "x1=np.array([1,0])\n",
    "#Apply mat2 once to x1\n",
    "x1_r1=#TODO\n",
    "\n",
    "#Apply mat2 twice to x1\n",
    "x1_r2=#TODO\n",
    "\n",
    "#Apply mat2 three times to x1\n",
    "x1_r3=#TODO\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "#this code will plot your rotated state vectors\n",
    "plt.close('all')\n",
    "plt.figure()\n",
    "plt.subplot(141)\n",
    "plt.arrow(0,0,x1[0],x1[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('Original Vector')\n",
    "plt.subplot(142)\n",
    "plt.arrow(0,0,x1_r1[0],x1_r1[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('1 Rotation')\n",
    "plt.subplot(143)\n",
    "plt.arrow(0,0,x1_r2[0],x1_r2[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('2 Rotations')\n",
    "plt.subplot(144)\n",
    "plt.arrow(0,0,x1_r3[0],x1_r3[1])\n",
    "plt.ylim([-2,2])\n",
    "plt.xlim([-2,2])\n",
    "plt.title('3 Rotations')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}