{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "id": "eXcJc7ELc2Bq"
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import linalg as la"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "id": "COF5AdaIsIvU"
   },
   "outputs": [],
   "source": [
    "# Let's start with a symmetric matrix.\n",
    "S = [[1, 2], [2, 1]]\n",
    "\n",
    "# Let's use NumPy to do the eigenvalue decomposition of S.\n",
    "l, U = la.eig(S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Ryi1YnVvsVTz",
    "outputId": "60fc0128-d890-48a6-ac7f-301fd7bb64f5"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 3., -1.]),\n",
       " array([[ 0.70710678, -0.70710678],\n",
       "        [ 0.70710678,  0.70710678]]))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l, U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-1.,  3.]),\n",
       " array([[-0.70710678,  0.70710678],\n",
       "        [ 0.70710678,  0.70710678]]))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# For Hermitian (symetric) matrices, Numpy has a more efficient version called 'eigh'\n",
    "# which also returns the eigenvalues in sorted order.\n",
    "l, U = la.eigh(S)\n",
    "l, U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "DsFFSR7OsWoL",
    "outputId": "6293dee7-c3a1-4b70-bee9-f5c5b4ebd2c6"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 3., -1.])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Show the vector of eigenvalues. We can see that S is not positive semi-definite, as it has a negative eigenvalue.\n",
    "l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "J8IcZ9zUsXjr",
    "outputId": "8a961922-667f-4014-bd4a-9c9b79bb6351"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.70710678, -0.70710678],\n",
       "       [ 0.70710678,  0.70710678]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Show the matrix of eigenvectors, one per column.\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "aV0_ZyWwsYWr",
    "outputId": "0c761880-fae4-429e-cfb2-96e66f2315e3"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2.12132034, 2.12132034])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute S x u1, in order to show that S x u1 = lambda1 x u1.\n",
    "S @ U[:,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "jnBquBh8smEq",
    "outputId": "af665c68-5622-42a6-a734-68b8f2db1f81"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2.12132034, 2.12132034])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute lambda1 x u1, it should be equal to S x u1 above.\n",
    "l[0] * U[:,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "knZ-8I8ksqg6",
    "outputId": "d7807066-720c-4242-e69d-5147db21874b"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9999999999999999"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Show that the norm of u1 is equal to 1.\n",
    "U[:,0].T @ U[:,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "3o3LjwBIs0qC",
    "outputId": "0191eec7-c3d0-4baf-ae12-539b12d4f2de"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Show that u1 is orthogonal to u2.\n",
    "U[:,0].T @ U[:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "3QAd_5tds4TC",
    "outputId": "f2610bef-e463-4c02-97ee-0ec3f5733352"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.12132034,  0.70710678],\n",
       "       [ 2.12132034, -0.70710678]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute S x U, to check that S x U = Lambda x U.\n",
    "S @ U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Oe8SwZSvtbjB",
    "outputId": "479e9c16-bf74-4f18-9484-a38b9b68aaf0"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.12132034,  0.70710678],\n",
       "       [ 2.12132034, -0.70710678]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute U x Lambda, it should be equal with S x U above.\n",
    "U @ np.diag(l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "z182bIPYtgDB",
    "outputId": "9d10b3d2-3fb5-4ec4-851b-002e7e556e8e"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3.,  0.],\n",
       "       [ 0., -1.]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create the Lambda matrix containing the eigenvalues on the diagonal.\n",
    "np.diag(l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "cGuGMCLYtnPJ",
    "outputId": "1da6b48c-c2c1-480a-85ca-cf24c33aadfe"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0.],\n",
       "       [0., 1.]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This shows that the eigenvectors are orthonormal.\n",
    "U.T @ U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "h8S3PsJvtwFQ"
   },
   "outputs": [],
   "source": []
  }
 ],
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