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     "text": [
      "Diagonal matrix A (non-negative elements):\n",
      " [[ 5  0  0  0  0  0]\n",
      " [ 0  3  0  0  0  0]\n",
      " [ 0  0  8  0  0  0]\n",
      " [ 0  0  0  1  0  0]\n",
      " [ 0  0  0  0  5  0]\n",
      " [ 0  0  0  0  0 12]] \n",
      "\n",
      "C1 (A + A.T)/2: \n",
      " [[ 5.  0.  0.  0.  0.  0.]\n",
      " [ 0.  3.  0.  0.  0.  0.]\n",
      " [ 0.  0.  8.  0.  0.  0.]\n",
      " [ 0.  0.  0.  1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  5.  0.]\n",
      " [ 0.  0.  0.  0.  0. 12.]] \n",
      "\n",
      "C2 (A.T@A):\n",
      " [[ 25   0   0   0   0   0]\n",
      " [  0   9   0   0   0   0]\n",
      " [  0   0  64   0   0   0]\n",
      " [  0   0   0   1   0   0]\n",
      " [  0   0   0   0  25   0]\n",
      " [  0   0   0   0   0 144]] \n",
      "\n",
      "C1 = sqrt(C2)\n",
      "\n",
      "A.T@A simply squares the diagonal elements of A because the transpose is the same as A.\n",
      "\n",
      "A.T + A doubles the diagonal elements of A and then divides them in half, so A is unchanged.\n",
      "\n",
      "Hence, C2 is simply the elements of C1 squared, or alternately C1 is the square root of C2.\n"
     ]
    }
   ],
   "source": [
    "# D. Scott\n",
    "# Linear Algebra: TCI\n",
    "# Code Challenge 6.3\n",
    "# Creation of symmetric diagonal matrices\n",
    "\n",
    "import random\n",
    "import numpy as np\n",
    "\n",
    "A = np.empty((6,6),dtype=int)\n",
    "\n",
    "# generate diagonal 6 x 6 matrix\n",
    "for i in range(0,6):\n",
    "    for j in range(0,6):\n",
    "        r_num = 0\n",
    "        while (r_num <= 0):\n",
    "            r_num = random.randint(-4,12)        \n",
    "        if (i == j):\n",
    "            A[i][j] = r_num\n",
    "        else:\n",
    "            A[i][j] = 0\n",
    "\n",
    "print(\"Diagonal matrix A (non-negative elements):\\n\",A,\"\\n\")\n",
    "\n",
    "C1 = (A + A.transpose())/2\n",
    "C2 = A.transpose()@A\n",
    "print(\"C1 (A + A.T)/2: \\n\",C1,\"\\n\")\n",
    "print(\"C2 (A.T@A):\\n\",C2,\"\\n\")\n",
    "print(\"C1 = sqrt(C2)\\n\")\n",
    "print(\"A.T@A simply squares the diagonal elements of A because the transpose is the same as A.\\n\")\n",
    "print(\"A.T + A doubles the diagonal elements of A and then divides them in half, so A is unchanged.\\n\")\n",
    "print(\"Hence, C2 is simply the elements of C1 squared, or alternately C1 is the square root of C2.\")"
   ]
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