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    "### Machine Learning (Background): Inner products\n",
    "$\\mathbf{u}\\cdot\\mathbf{v}=\\sum_{i=1}^n u_i v_i \\,\\rightarrow \\lVert \\mathbf{u} \\rVert_{canonical}=\\sqrt{\\mathbf{u}\\cdot\\mathbf{u}}$ <br>\n",
    "$<A,B>_F=tr(A^HB) \\, \\rightarrow\\lVert A \\rVert_F=\\sqrt{<A,A>_F}$\n",
    "###### by Hamed Shah-Hosseini\n",
    "Explanation at: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/Background-Knowledge\n",
    "<br>Explanation in Persian: https://www.instagram.com/words.persian\n",
    "<br>Code that: https://github.com/ostad-ai/Machine-Learning"
   ]
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  {
   "cell_type": "code",
   "execution_count": 98,
   "id": "ddd68127",
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   "source": [
    "# importing the required module\n",
    "# درون‌بَری سنجانه نیازداشته\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "id": "b12e168b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vector 1: [2 1 1 3]\n",
      "vector 2: [1 2 4 3]\n",
      "dot product:  17\n",
      "1-norm of vector 1: 7.0\n",
      "1-norm of vector 2 10.0\n",
      "2-norm of vector 1: 3.872983346207417\n",
      "2-norm of vector 2 5.477225575051661\n",
      "canonical norm of vector 1: 3.872983346207417\n",
      "canonical norm of vector 2 5.477225575051661\n"
     ]
    }
   ],
   "source": [
    "# dot product and p-norms, example\n",
    "# فرآورد خجک و پ-هنجارها: نمونه\n",
    "vec1=np.random.randint(1,5,4)\n",
    "vec2=np.random.randint(1,5,4)\n",
    "print('vector 1:',vec1)\n",
    "print('vector 2:',vec2)\n",
    "print('dot product: ',np.dot(vec1,vec2))\n",
    "print('1-norm of vector 1:',np.linalg.norm(vec1,ord=1))\n",
    "print('1-norm of vector 2',np.linalg.norm(vec2,ord=1))\n",
    "print('2-norm of vector 1:',np.linalg.norm(vec1,ord=2))\n",
    "print('2-norm of vector 2',np.linalg.norm(vec2,ord=2))\n",
    "print('canonical norm of vector 1:',np.sqrt(np.dot(vec1,vec1)))\n",
    "print('canonical norm of vector 2',np.sqrt(np.dot(vec2,vec2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "id": "1a0b718b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The matrix:\n",
      " [[3 2 2]\n",
      " [3 1 4]\n",
      " [1 3 2]]\n",
      "---------------------\n",
      "Eigenvalues of matrix: [ 6.92434399  1.21506597 -2.13940996]\n",
      "Trace of matrix: 6\n",
      "sum of eigenvalues: 5.999999999999998\n",
      "-------------\n",
      "Determinant of matrix:  -18.000000000000004\n",
      "Product of eigenvalues:  -18.0\n",
      "------------------\n",
      "Frobenius norm of matrix: 7.54983443527075\n",
      "Norm using Frobenius inner product: 7.54983443527075\n"
     ]
    }
   ],
   "source": [
    "# Example for trace, determinant, and norm of a matrix\n",
    "# نمونه برای رآس، بَریهنده، و هنجار یک ماتکدان\n",
    "M=np.random.randint(1,5,(3,3))\n",
    "eigs=np.linalg.eigvals(M)\n",
    "detM=np.linalg.det(M)\n",
    "print('The matrix:\\n',M)\n",
    "print('---------------------')\n",
    "print('Eigenvalues of matrix:',eigs)\n",
    "print('Trace of matrix:',np.trace(M))\n",
    "print('sum of eigenvalues:',np.sum(eigs))\n",
    "print('-------------')\n",
    "print('Determinant of matrix: ',detM)\n",
    "print('Product of eigenvalues: ',np.prod(eigs))\n",
    "print('------------------')\n",
    "print('Frobenius norm of matrix:',np.linalg.norm(M,ord='fro'))\n",
    "print('Norm using Frobenius inner product:',np.sqrt(np.trace(M.T@M)))"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "id": "0df1d5e9",
   "metadata": {},
   "outputs": [],
   "source": []
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