9. Linear Algebra.ipynb

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   "source": [
    "#### Linear Algebra\n",
    "Work in progress. Basic linear algebra to be included here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Simple Array Operations"
   ]
  },
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   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
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   "execution_count": 4,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 2.],\n",
       "       [3., 4.]])"
      ]
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     "execution_count": 4,
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   "source": [
    "a = np.array([[1.0, 2.0], [3.0, 4.0]])\n",
    "a"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 3.],\n",
       "       [2., 4.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.transpose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2. ,  1. ],\n",
       "       [ 1.5, -0.5]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.inv(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0.],\n",
       "       [0., 1.]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
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   "source": [
    "u = np.eye(2) # unit 2x2 matrix; \"eye\" represents \"I\"\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-1.,  0.],\n",
       "       [ 0., -1.]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j = np.array([[0.0, -1.0], [1.0, 0.0]])\n",
    "j @ j        # matrix product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.trace(u)  # trace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-3.],\n",
       "       [ 4.]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = np.array([[5.], [7.]])\n",
    "np.linalg.solve(a, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0.+1.j, 0.-1.j]),\n",
       " array([[0.70710678+0.j        , 0.70710678-0.j        ],\n",
       "        [0.        -0.70710678j, 0.        +0.70710678j]]))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eig(j)"
   ]
  }
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