demo3_Eigenvalues_of_matrix.ipynb

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   "cell_type": "markdown",
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    "# Finding Eigenvalues of a $2\\times2$ Matrix\n",
    "\n",
    "\n",
    "## 1. Definition of Eigenvalues and Eigenvectors\n",
    "Consider the linear transformation of $n$-dimensional vectors $v$ defined by an $n\\times n$ matrix $A$,\n",
    "\\begin{equation*}\n",
    "Av = w\n",
    "\\end{equation*}\n",
    "\n",
    "or\n",
    "\n",
    "\\begin{equation*}\n",
    "\\left[\\begin{array}{cccc}\n",
    "A_{11} & A_{12} & \\cdots & A_{1 n} \\\\\n",
    "A_{21} & A_{22} & \\cdots & A_{2 n} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "A_{n 1} & A_{n 2} & \\cdots & A_{n n}\n",
    "\\end{array}\\right]\\left[\\begin{array}{c}\n",
    "v_{1} \\\\\n",
    "v_{2} \\\\\n",
    "\\vdots \\\\\n",
    "v_{n}\n",
    "\\end{array}\\right]=\\left[\\begin{array}{c}\n",
    "w_{1} \\\\\n",
    "w_{2} \\\\\n",
    "\\vdots \\\\\n",
    "w_{n}\n",
    "\\end{array}\\right]\n",
    "\\end{equation*}\n",
    "\n",
    "\n",
    "$\\lambda$ is the \\textbf{eigenvalues}, and $v$ is the \\textbf{eigenvectors} of the matrix $A$, if\n",
    "\\begin{equation*}\n",
    "Av = w = \\lambda v\n",
    "\\end{equation*}\n",
    "\n",
    "E.g. for a $2\\times2$ matrix $A$,\n",
    "$$ A =\n",
    "\\begin{bmatrix}\n",
    "0 & 1 \\\\ \n",
    "-2 & -3\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "The two sets of \\textbf{eigenvalues} and \\textbf{eigenvectors} of the matrix are:\n",
    "$$ \\lambda_1 = -1, \\lambda_2 = -2,\n",
    "v_1 = \n",
    "\\begin{bmatrix}\n",
    " 1 \\\\ -1\n",
    "\\end{bmatrix},\n",
    "v_2 = \n",
    "\\begin{bmatrix}\n",
    " 1 \\\\ -2\n",
    "\\end{bmatrix},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Solution for finding the eigenvalues\n",
    "\n",
    "Rewritten the function above,\n",
    "\n",
    "\\begin{equation*}\n",
    "Av - \\lambda v = (A-\\lambda I)v = 0 \\\\\n",
    "|A-\\lambda  I| = 0\n",
    "\\end{equation*}\n",
    "\n",
    "E.g. for the given matrix $A$\n",
    "\n",
    "$$ |A-\\lambda I| =\n",
    "|\\begin{bmatrix}\n",
    "0 & 1 \\\\ -2 & -3\n",
    "\\end{bmatrix}\n",
    "-\\lambda \n",
    "\\begin{bmatrix}\n",
    "1 & 0 \\\\ 0 & 1\n",
    "\\end{bmatrix}|\n",
    "= \n",
    "|\\begin{bmatrix}\n",
    "-\\lambda & 1 \\\\ -2 & -3-\\lambda\n",
    "\\end{bmatrix}|\n",
    "=\\lambda^2 + 3\\lambda + 2 = 0\n",
    "$$\n",
    "\n",
    "According to the function above, we then get $\\lambda_1=-1$ and $\\lambda_2=-2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Solution for finding the eigenvectors\n",
    "\n",
    "$$\n",
    "(A-\\lambda I)v = \n",
    "\\begin{bmatrix}\n",
    "-\\lambda & 1 \\\\ -2 & -3-\\lambda\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "v_1 \\\\v_2\n",
    "\\end{bmatrix}\n",
    "=0\n",
    "$$\n",
    "\n",
    "For the first eigenvalue $\\lambda_1=-1$, we have\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "1 & 1 \\\\ -2 & -2 \n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "v_{1,1} \\\\ v_{1, 2}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "v_{1,1} + v_{1, 2} \\\\ -2v_{1,1} - 2v_{1,2}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "0 \\\\ 0\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Hence, the first eigenvector $v_1$ can be $[1 -1]^T$.\n",
    "\n",
    "\n",
    "Likewise, we can calculate the second eigenvector $v_2$ = $[1 -2]^T$ for the second eigenvalue $\\lambda_2=-2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "solution2": "shown",
    "solution2_first": true
   },
   "source": [
    "### Exercise: Calculate the eigenvalues and eigenvectors using Numpy\n",
    "\n",
    "\\textbf{Task}:\n",
    "\\begin{enumerate}\n",
    "\\item Define the matrix A and convert it to a numpy array\n",
    "\\item Find the eigenvalues and eigenvectors via the function, (values, vectors) = np.linalg.eig(matrix)\n",
    "\\item Print the eigenvalues and eigenvectors one by one. (Format)\n",
    "\\item Plot the Arrows for the two eigenvectors using \\textit{Matploblib.pyplot} library\n",
    "\\end{enumerate}\n",
    "\n",
    "\n",
    "*Tips: \n",
    "np.linalg.eig() can automatically calculate the results for you. \n",
    "\n",
    "\\textbf{Print format}: \n",
    "\n",
    "'Eigenvalue: -1, EigenVector: [1.00; -1.00]'\n",
    "\n",
    "'Eigenvalue: -2, EigenVector: [1.00; -2.00]'\n",
    "\n",
    "Keep 2 decimal space\n",
    "\n",
    "\\textbf{Plotting Arrow}:\n",
    "import matplotlib.pyplot as plt\n",
    "plt.arrow(start_x, start_y, offset_x, offset_y, head_width, head_length, color)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "solution2": "shown"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalue:-1.00, EigenVector:[0.71; -0.71]\n",
      "Eigenvalue:-2.00, EigenVector:[-0.45; 0.89]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "A = np.array([[0, 1],[-2, -3]])\n",
    "e_val, e_vec = np.linalg.eig(A)\n",
    "\n",
    "for i in range(len(A)):\n",
    "    print('Eigenvalue:{:.2f}, EigenVector:[{:.2f}; {:.2f}]'.format(e_val[i], e_vec[0, i], e_vec[1, i]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "solution2": "shown"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x129780d30>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.axes()\n",
    "plt.xlim([-1, 1])\n",
    "plt.ylim([-1, 1])\n",
    "arrow1 = plt.arrow(0, 0, e_vec[0, 0], e_vec[1, 0], head_width=0.05, head_length=0.1, color='blue')\n",
    "arrow2 = plt.arrow(0, 0, e_vec[1, 0], e_vec[1, 1], head_width=0.05, head_length=0.1, color='green')\n",
    "\n",
    "plt.legend([arrow1,arrow2], ['v1', 'v2'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda env:new_env] *",
   "language": "python",
   "name": "conda-env-new_env-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.13"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": false,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {
    "height": "calc(100% - 180px)",
    "left": "10px",
    "top": "150px",
    "width": "345.188px"
   },
   "toc_section_display": true,
   "toc_window_display": true
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}