asymmetry_index.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       " 0.427566   0.926303  0.768389   0.252602    0.068628\n",
       " 0.963839   0.626034  0.621404   0.720021    0.927462\n",
       " 0.462503   0.263531  0.0798756  0.996313    0.819039\n",
       " 0.0877163  0.670902  0.634118   0.00941859  0.664718\n",
       " 0.438494   0.45174   0.969794   0.0982301   0.551076"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = rand(5,5)\n",
    "B = rand(5,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       " 0.197615  0.544369  0.496345  0.398907  0.488414\n",
       " 0.544369  0.267222  0.367176  0.354835  0.40733\n",
       " 0.496345  0.367176  0.380197  0.298756  0.624173\n",
       " 0.398907  0.354835  0.298756  0.095137  0.319696\n",
       " 0.488414  0.40733   0.624173  0.319696  0.132158"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A_sym = (A+A')/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       "  0.0       -0.425335  -0.515879   0.458266   0.465366\n",
       "  0.425335   0.0       -0.493954   0.554321   0.362223\n",
       "  0.515879   0.493954   0.0        0.518249  -0.613547\n",
       " -0.458266  -0.554321  -0.518249   0.0        0.394691\n",
       " -0.465366  -0.362223   0.613547  -0.394691   0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A_skw = (A-A')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "asym_idx (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "function asym_idx(A) \n",
    "    A_tilde = (A-A') / 2\n",
    "    a_i = norm(A_tilde) / norm(A)\n",
    "    return a_i\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "skw_mat (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function skw_mat(A)\n",
    "    @assert size(A)[1] == size(A)[2] \"Matrix must be square\"\n",
    "    @assert round.(A-A', digits=2) == zeros(size(A)[1], size(A)[2]) \"Matrix must be symmetric\"\n",
    "    \n",
    "    skw_mat = copy(A)\n",
    "\n",
    "    for i in 1:size(A)[1]\n",
    "        for j in 1:size(A)[1]\n",
    "            if i < j\n",
    "                skw_mat[i,j] = -A[i,j]\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    return skw_mat\n",
    "end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4690613478086576"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Random Matrix\n",
    "ai_1 = asym_idx(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Symmetric Matrix\n",
    "ai_2 = asym_idx(A_sym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ai_2 = asym_idx(A_skw)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "skew_blend (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Formula to blend between symmetric and assymetric matrices\n",
    "function skew_blend(M_sym, M_skw, proportion)\n",
    "\n",
    "    @assert 0 <= proportion <= 1\n",
    "\n",
    "    M = (1 - proportion) * M_sym + proportion * M_skw\n",
    "\n",
    "    return M\n",
    "\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       "  0.0       -0.425335  -0.515879   0.458266   0.465366\n",
       "  0.425335   0.0       -0.493954   0.554321   0.362223\n",
       "  0.515879   0.493954   0.0        0.518249  -0.613547\n",
       " -0.458266  -0.554321  -0.518249   0.0        0.394691\n",
       " -0.465366  -0.362223   0.613547  -0.394691   0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A_skw"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       "  0.0494036  -0.182909   -0.262823    0.443426    0.471128\n",
       "  0.455094    0.0668054  -0.278672    0.504449    0.3735\n",
       "  0.510995    0.46226     0.0950491   0.463375   -0.304117\n",
       " -0.243972   -0.327032   -0.313997    0.0237843   0.375942\n",
       " -0.226921   -0.169835    0.616204   -0.216094    0.0330394"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = skew_blend(A_sym, A_skw, .75);\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "asym_idx(A_skw)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9541245457541365"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "asym_idx(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Test on A matrix\n",
    "\n",
    "proportion_list = zeros(100)\n",
    "step = .1\n",
    "\n",
    "for p = 1:100\n",
    "    X = skew_blend(A_sym, A_skw, p/100);\n",
    "    idx = asym_idx(X)\n",
    "    proportion_list[p] = idx\n",
    "end\n",
    "\n",
    "using Plots\n",
    "plot(proportion_list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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zbD3b0+ydhPvb+Kw0j7S7o1RYf5q7VlbztSSb2SX66UxBhXJwXEMQAsD4UW/maVp2TQHzXTV7X2vv3AmXWanMEEO/Oyz5tovKYgT6bIX6jlJ8To5zeIEBYPyo6qFCA12RL2yp8a4I97Xx2WnegXf7RKHBzC/J8ftJOD2FoRwc97DWKACMH9UmXmBgCzNYZQ9vtlCGru+4kasZZeu9z7+jVOiw8RSN93GIKqgIAWD8qDJSoYHUAl2SI2yt628d/aZVoRwkokwdPTVX5XscogqCEADGj2ojl3cHvDKPubeO7u+bSg/gC0EIAONHlZEXGBgRXZUvuE+i2NeqsAEhgAxBCADjR7WRCg1ERNl6yo1jX/etmr2/jVARgj8IQgAYP6r7KkIiuiKPbamViKjJQlax/ziAFwQhAIwT7TZSMUqI6b15Zd8kCnmkDGIQ/EEQAkCItduowRyE53GNlJEtymSnuniLlfa38XJ0EIJ/CEIACLG/nZDW7BdH/jxVRo/2zxh5EkWt5LumDIA7BCEAhFijhVcbFfaLGKqqvpEyLlfksQ9r+T7MnYABIQgBIMSaLVRlDMLz1Bi9R8Rcmc8+qJbarXxiAoIQ/MISawAQYk3BqwjnpnscyYtjeXEsRUPYTRAGgCAEgBBrtlC3gzrtlBQ7ouepNvECg3cr14oC5lDYrx6gH4IQAEKs2crTtVRt5EkpIyrcqnp4gcH74BPnq3gQqk0Yz9BHCAChJHLqsNH5aax6ZN2ENpE67ZSl845SrYp0+MIPA0IQAkAotVgpWUMT4tkIuwmrjTwvjqEvEIYBQQgAodRk4Zk6VmBgNaYRBWGVkXzbRQECgSAEgFBqtlCGlgriaIRNo9Umj2VlAAKHIASAUHJVhNUjrAiVRsoABAJBCACh1GyhDB0VGKiqZ0TPU20iVIQwPAhCAAgluSLM1bNmKx/JhL+qHmy0BMOEIASAUGq2UoaO1AJlaFm9efito9Um74VGAQKEIASAUJIrQiIqMAx/vIzEqc7E8+JQEcJwIAgBIJTkUaNEVGAY/lTCRgtPiiWtKpgXBtEDQQgAodRkoUwdEVGhgapNw3ySaiMVxqMchGFCEAJAKLVYebqWEVF+HKsZbkVYZeQFaBeF4UIQAkDIdNpJI/SuBVpgYFXDD0KMlIHhQxACQMg0WXimvreSG8lgmWqfLXkBAocgBICQae7rIKSRVYTVqAhhBBCEABAyTRaeoe2t5JJiiRF12ofzPFWoCGEEhrZP16lTpwRBmDhxou9dFovFarW6H0lKSmKMmUwmu733rc0YS0pKGva1AsA40+RWEVLfDIphbM9bZcSK2zB8gQah0WhcsWJFfX29JEklJSXvvPOOTqdzP+FPf/rTb37zG/lnu90uSVJHR4dGo7n77rvfffddjUZDRElJSWfOnAnuPwAAwt+P94j3TRZKE72zqsXK07X9NwsNVGXkM3yC0C5RrP/Wq24HSZySNUG7Wog2gTaNPvfccyqV6sSJEydOnDCZTGvXrvU64bHHHmvvs3r16uuvv14OPyL67W9/Kx9HCgJEp801fE+LQv9fk4Uy3faULzAo7FN/qotnb3Dsb/PbfYhVRmGEAg3CV1999bvf/a5KpYqJiVm9evVrr73m70y73f7qq69+5zvfcR1xOp1NTU0jvVIAiFgNZl7RpZBk8tYTLorb835Ux3Pj2NVbxVNKz0BYZRRGLNAgrKqqKi4uln8uLi6uqqryd+bbb78dHx+/dOlS15Ff/OIX5eXlSUlJzzzzzAC/QhTFgwcPftOnuro6wGsDgNFWY+IH24c5pNPspC47nepWuMu10KhMcQbF9jr+05nCL+cIyz8Ua5X2LERFCCMUaB+h2WzWanvb8nU6ndHod77Piy++uHr1akHojdjf/OY3GzZsEARh165dl19++YwZM9wz0p3FYnnwwQdVqt7lApcsWbJmzRrXvQP8RhgbJpOJ8xFtnQojZDabRVFkLAQf+i8eV+9tEzZeOJwxnWeNjJHmRLuzp8fsdVejWRMnWXt6et9XaUw426V2P03ktKNB87tye4aWN3arl30gfXiJPVXj8T483aHOjKGeHucwrm2ozGaz0+l0fb7B2LNYLHa73ZUUg9JqtTExMQOfE2gQZmZmtre3yz+3t7dnZWUpnlZTU7Njxw73HsT8/Hz5h4ULF1511VWffPKJvyA0GAw7duxISUnxdw3x8fEBXi2MBsaYwYAWqFASBEGv14ckCGtt4t52aXh/g90mPjVZrDQKvg9vtjpK0gzxsb03JzFeZxHdT9vTzPMNYkm6gYh+OofMTLz239odV6uTYvufpMEuzs1i8fEew/dGiSAIOp0OQRhCKpVKo9EEHoSBCDQIzz///C+//PLSSy8lol27dp1//vmKp73wwguXXnppQUGB4r2NjY3z588f3oUCQAid6eHNFjrXw4uGvrZ1o4VPTmJNFt5gpmx9/3GrSHaJEtwizbU9b0xf0HzSwC/N6f+Na2ar2m3ilR8652Uwi5O67GQT6d+N0gNThjYTDMBdoO+eH/3oRzfeeOOUKVOcTufzzz+/ZcsW+Xhpaekrr7wyb948IpIk6eWXX/7tb3/repTD4XjooYcuu+wyrVa7adOmI0eObNy4Mej/BgAYbWd6aEYK29MynCCU868skVV082x9/8ObLTxd61HeqgXK1LE6U/9v+bhOemiax3f/Zxeo/n5SsjhJr6aEGNKq6UfT1PMy0EcIwxdoEF588cUvvvjiSy+9xBj75z//6SrsrrjiCldjZm1t7VVXXXXNNde4HqVSqVJSUl588UWn01lWVrZv377c3Nzg/gMAYLQ5JGow8++Wq75q4d8qHvLD6808R896Eqmiiy/O6k+sRgtl+TRnFhioxkRF8UREVpH2tvDF2R4hJzC6dzJaJiGYhtCesGrVqlWrVnkdfPbZZ10/FxQUPP/88+73CoLw5JNPjuT6ACDkzvXwXD27MIv9/GtpGA9vMNPkJJI4q+j2GOTiNXdCVhAnb8/LiOjzRj4jhSUMMtABYKTwxQoABnHWSMUJNCeNHWrnDj9RaBP9PrzRwjN1rCyRTnZ6HG+28gydd5NmoYGq+kaIb6+X3DsIAUYJghAABlHZzYvjmSGGiuOZ4mzCym5e+rrf2QuNZsrWU2miUkWo9T45321O/fZ6fmkuPqNg1OFNBgCDONPNi+MZEc3LYHuaFYLwX1W8xsR7HMoPrzPzHD0rTWBnerjo9uhGC8/Sexd88rrbRNRpp5OdfF46KkIYdQhCABjEmR4qTiAimpeuHITvVkkqRtVKuwk6JOq2U6qG9GpK07Aat3NaLJTuUxEWxPU2jX5aLy3MZJpgzhYDUIYgBIBBVHbzEldF6LN2drOFDrXzi7IU1ssmecdBHRMYEVFpIlW4LbTWaOFZvn2E8b0V4cf1fBnaRWFM4H0GME4caOMbTg9nVOegzvbw4gRGRFOSWJOFt9s87n2/RlqeJ0xKZOeUKkL3ORJlicx94ewWK6X7jBpNiCGBUYeNttfxSzBSBsYEghBgnPisgX/vc9F394YRarGSWiB5STOB0ew09pVnUfhOFb+2kBXGsyqlIGww90+iL03wCEKvFbddCgxsVzNvt/GZQ9+hF2AYEIQA40SrjevV9NO9QS4KXSNlZF7dhCYnfdYgXZkvuE97cOe+rFqZ28BRkVOHjdKUdtMtNNBLp6SLcwQBOQhjAkEIME60WumRGarPGviXSuNZhu1MX7uobF4G29PSn7Uf1UoXpLOkWCo0sKoexYqwPwhLE+lUV//VJsWSWukTqMDA3qvCDEIYOwhCgHGixUpF8fSrucIDu0QpeFFY2U0lbptGLMgQvmru345rUxW/tlAgz4nw7uT11eSfi+NZnYnbJSL/7aJElB/H7BIty0UQwhhBEAJEjEf2qb7yGbTp0mblaVr2HxMFtUAbKoPWQHrWsyLM0JEhhp3u4kTklGhzjXRNISOiHD1rs3Hf9WUaLZTZNyImRqC8OHamm5M8UsZn7oSswEBF8ax46Kt7AwwPghAgYuxpFY51+A3CFiulaYgR/XG+6md7JVOQ9qmt7OFemeSaRPF5E58Qz/LiGBEJjHL1zHeoTqPZY8eJskSSuwmbLDzTZza97JIc4el5+GiCsYN3G0DEaLCwZqvfe1usPF3HiGh+BluSzX590P/qn0NxpptKEjyOuMbLbKqS5HZRWaGBfKcS1pkpx20PwtJEJncTNpoVtp6QZenouiJ8NMHYwbsNIDJInJqt1OonCDlRh41S+wZh/nqu8Ofj0jml0StDYhOpxcrlms/FVRG+W8VXFfbfVRjvPZWQE7V49gWWJrCKLk5+VtwGCAkEIUBkaLGSUyJ/FWGnjfTq/o3dc+PYD6eoHt830p7Cc0aeb2Aqz8A6P5Ud6+B7mrmK0dTk/vuKfAaOtlopMbb/qohoUt+cesUVtwFCAkEIEBkazHJ+KBd5rVaervXIq5uLvWe+D8OZbiqO9z6oU9PkJPb4PtG9HCSlgaONZu9ltUsT6VQ3EVGzhftuRggQEghCgMhQb6Y0DfdXEbbaKM2zwMrSsSY/qRm4Mz4jZWTzMtjWWr6q0OMDxHdxmQYLZXumXX4c67Bxk5OarZShRdMohAUEIUBkaDDz6cm82aJ8b6uVewVhkoYszoH2yw1EZbfH3AmXeeksQ0cLMwepCBs8h4wSkcCoOJ5VdPEmt2kVAKGlDvUFAEBAGsw0PYnv8tPa2WKlNM8CixFl6FiThRcYhl94nemhC7MUjq8sEFK13n2HeXGs3sxFTq7j7svKuMhLbzdbMFgGwgUqQoDIUG/mxfEUK1CXXeHeFqWxJ5k6avRTQQaospuXKFWEyRpake99PFagNC2rN/dHtfuyMi6lifR1K48VSI/v4RAeEIQA4cIh0UXvO/116zWYKUMjZeiY4niZNhtP0XhHzgi7CTnROSOfMJQVXooMVNXTf7PRrND+WZbIvmhCOQhhBEEIEC4azPzzRr+9gA0WnqWjDK3yDIpWpb39MnXUNIKKsNlCOhUlxAzhIYUGj6mEDRbvPkIiKk1gX7dgyCiEEQQhQLioMxMRKe7qR0T1JsrRc38VYYuV+y7dmaWnRvPwr8dfu+gAvBaXqTd5LCsjK01kdon8rbgNMPYQhADhos7EyU8QcnkpFi1l6PxWhGk+sxEyR9Y06m/uxAC8ZlA0WrznERJRlo4SYzFkFMIIghAgXNSbiYjO9Sjc1WKhhBiKFShDS4ptp/KK216yRjZYprKbihMGP81doaE/CLsdpGYUpzQiZmICQxBC+EAQAoQLeYylYkXoKq3SNNRiVRosY+WpShVh4wgqwrPDqAgN/UHe4LOsjMvkJIamUQgfCEKAcFFvooWZTHGl7Ia+PRwydAoVoUMik5OSlSrCkQyWGU7TqIHVmHq37W1UmkQoe3qe6o5SfPhAuMB7ESBc1Jr4wkxWbVK4q65vQl66lvlWhK1WStWQb2Rl6lmjefgVYaXPBkyD0qtJr6YWC5GfSYSyDJ1ykylASCAIAcJFvZkW+a8Is/1XhC1W7jtShogSYkjiNLwdeq0itdv8JtkAivq6CRWXlQEIQwhCgHBRb+bnJTEVo3ab912uRTszdKxZqSJM87Olkb/pFoM608OL4pkw9I4818DRBjPPQkcgRAIEIUBY6HYQJ4qP8Rh46eLqI0zXUoeNRM/7W/1UhDSCgaOV3dx3A6ZAFBronJFI3noCFSFEAgQhQFioN/FcPSOiIp/NjIiovq8iVDFKjKU2z6mErVbynU0vy9QNs5vwTDcNdTa9rLBve17frScAwhOCECAs1PfVfO4zEFwaLZTVN/Euw2e8TJuNUv0EYZZ+mANHh7rKqEtR32ZMTW7XDBDOEIQAYaHOzHPjGJFC0ygnanSrrnwXl2mx8jSfFbdlmToa3lRCVzAPVUHf9Q8wahQgrCAIAcJCnYly44iUtrdtt1GcmrSq3pvpWu/xL4orbsuydGx4FWGTZZhDXQoN7FwPt4lkcU/JL94AACAASURBVFKSz9RGgDCEIAQIC676qSjeewZFvcmjs813BoW/6RM0gg0oFHdQCkSyhhij4508S89QD0JEQBAChIV6M+X29hF6N402WDxaKX3n1LcqLTQqyxrunPompfWyA1RgYLubeTY6CCFCIAgBwkKdiefEMSJK05JDom5H/12DVoQDzCMc3ib1NpEsIiXGDvmBskID7W7GkFGIGAhCgLDgqgiJKD+OVXtsb+sxIc93b94B5hFmD2snpkYLz9INv2GzUK4IMYkQIgSCECD0uGdTZFG8xwyKRs8lWrwWizE5SWCk97N0p05NKkY9DoW7djZyyU9ENltoJDvIF8WzU10c+0tApEAQAoSea7tBWVE8O+deEZopJ67/ZK/pEy0Wv+WgTLGbUOJ05YfOym7lJGwc7pBRWaGBOGFZGYgYCEKA0Ksz87y4/uAp8GwarfOckJehZS1uFWGL/2VlZIoDR88ZuclJdWblhzSaKWsEMVZoYESESYQQKRCEAGPnYDtfe1LyPV7vuVGDT9OoxxItSRoyO8km9t5ss1HqgNP1spS25z3WQURUa1KuCJsslDFguA5MDkIsKwORAkEIMHaeOyb97YRSEJp6l5WRFRk8m0YtHiMwGVGa2wyKVitPH7AZU7EiPNrBGVGd0t6HNLK5E0SUoSO9mnLiUBFCZEAQAowRp0SbzknHOrjoU4bVmbn7TMFCt3W3O+2kEbzHwrjPoBhg7oRMsSI82sHLU1mdnymGTZZhzqaXMaI9q9SoCCFSIAgBxsiOBj4hnuXEsZNd3vFTZ6JctwosU0c9DjI75bsUVux0n0HRYuXpAw6WUawIj3Tw5XnMX0XYYOHZIxvzOS0Z5SBEDAQhwBh586x04wShPIUdaPMJQrNH0yhzm0qouM+7+wyKwStCPTV6DoqROJ3q4pfkCKNUEQJEFgQhwFgQOW2qkq4vYjNT2UGfIKw3eW/14Fp6u8GiVBHq3CtCv+uryTJ95tSf6eEZOjY5yW8fYaN5RH2EAJEFQQgwFj5r4HlxrCSBlacOXhGS21TCepNCRZjuNoNigGVlZL6b1B/t4FOTKVvHmi0KHZYWJ4mc4mMC+4cBRD4EIcBYkNtFiag8hfZ7BqFdoh6Hd/Nm/z7vFoVFOzN01NJXEQ6wB5NMrgjdf+XRDpqazNQCpWrJdwG2RgsWhYHogiAEGHUSp01V0o0TGBHlxjHGqMGt005eU9srefqbRpU2yE3XUrN7RehnV16ZRkV6NXXa+o8c7eBTkxkR5eoVxss0Ymd5iDIIQoBRt7ORZ+nYxITeuJrhOV6mTinqitz2eVeoCLVM7iOUOHXaKWWw/W+9ugmPdvCpSb2p7DtepgkVIUQZBCHAqHO1i8rKU9iB9v74qTfzXJ+oK4ync30VodKo0d55hO02io8h9WB/x+7dhCKnim4+qTcIFcbLjHB9NYCIgyAEGF0Sp7fP8euL+qPOa+BoncljTW1Zjp61WrlNpEalijBTx5qtnAIYKeM637XudmU3z9azODVRb9Oob0WIuRMQXRCEAKPriyaepqXJSf1x5TVwtN6sMEFCxShHz450cLVAcT5bLOnVJBD1OKh1sBW3ZVn6/jn1rg5CkitCn3W30TQK0QZBCDC6vNpFiWhyIqsxcWPfHoHuW/K6KzLQl/73eZfn1LfaAq0IXX2ERztoalLvccWKEINlINogCAFGESf61zkujxd1UQt0XhI70tGbQHUmrrg+dWE8293Ms/1kkjyDosUyyLIysky3PsKjnf0VYV4c1fr0EaIihGiDIAQYRV828cRYOi/JO1fcW0f9VYSFBtrdrJyR1DeDotUWUBBmeVSE7k2jCqNGMVgGog2CEGAUbamVVhUqJNnMFHaw3RWEyu2fRQZW2c39tVLKMyjarDx1wEmEMteoUadEld18UmLvQ+JjiBF1OzxObrbyjACaWwHGDQQhwCg60UnTUxRCxVURdtlJIOX1zArjB9rnXZ5BMej29DLXBhSnu3luHNO5jb7JjfPoJuxxEBEZsL4aRBMEIcAoqujipQlKFWEqO9LBJa6wyqhLoYGIFCYRytK1rMXKA58+0WLhEu+fSu+Sq/eYSthk4VnoIIQogyAEGC2cqLKHT0xUyJWEGMrQsopu7rvvhEt+HJMnUSjeK1eEAU6fUAuUEEvtNjraSVOTPe7K8+wmbDRjEiFEHQQhwGipN3GDmhL8NDPKraOKuyzJYgQqMLA8n7n2sgwda7byNhulBhCE1LdP/bEOPsVzy9xcz4GjzVYMGYWogyAEGC0V3VSqVA7K5PVlFJeVcdl3nbpEqWWV5E3qLdRiCahplPq6Cd2HjMq8phJiWRmIQghCgNHir4NQVp5CB9q54kKjLkmxfp88Q0e1Jm4VKdH/Oe6y9KzGyCt7+oeMyrwWl8GWvBCFEIQAo+VUFx+gIpSbRmtNlOu/IhxAupZ12ChVSwGmVqaOdjbygjimVXkc9xo1imVlIAohCAFGS0U3lSb4vbfAwGwiHWzz20c4sBiBkjQUYLsoEWXp2CcN3u2ipNQ0moEghCiDIAQYLRVdvMx/RUhEM1PYmR4+vIqQiDK0LJAho7JMHZ3r4V5DRuXj7TZySL03Gy08G4NlIMogCAFGhcTprJH7G+oiK09lAqNhz9vL0A2lItQzIvKtCAVGmTrW0DeDAtMnIAohCAFGRbWRp2mY3mcHJXczU1mGlmKG+1eYoWOBLDQqk+PNNwjJc7wMpk9AFEIQAoyKim4qTRzknAUZ7IKM4f8NpmspVRPoyVk6FiOQYlOta7xMt4NiGOkGDG+A8QdveYBRMfDcCVlZInvnMtXA5wzgP0qEgStOd5k6+nSFOlYpdl2rrGHuBEQnBCHAqDjVPchImZG7MGtoz78oU/l812ZMmDsB0QlNowCjoqKLD9o0GiZQEUKUQxACjIqKbhq0aTRMuCpCrK8G0SnQIOzu7r7rrrtycnJmz569detW3xM2bNgwx01NTY18vLa2duXKldnZ2YsXLz5w4EDQLhwgjDklqjHyCfGREYR5cb0VYZMFQ0YhGgXaR/joo4+2t7cfPHhw165dN9100+nTpzMyMtxPaGpqys7OfuKJJ+SbrnvvuOOOadOmrVu3bv369ddcc82ZM2fUanRMwjh3tofn6Jlm+ONgxlSuvr+PcEHGoKcDjDcBVYQWi+WVV15Zs2ZNenr6qlWrFi1atH79et/T0tLSZvfRaDREVFFRsWvXrl/+8pfJyck/+tGP1Gr1li1bgvwvAAg/pwKYOxE+dGrSqqjdhooQolRAQVhdXW2xWKZPny7fLC8vP378uO9p27ZtKy8vv/LKKzdt2iQfOXHiRHFxcUJCguuBx44dC8ZlA4S1QRdXCzfyiqONZsrys0swwDgWUCtlW1ubwWAQhN7UTEpK8s2zRYsWTZ8+PScnZ8+ePbfffvuGDRuuueaatra2+Ph41zlJSUmtra3+fkt3d3dxcTFjvR8fy5cv//vf/+6612g0BvhPglFiMpk454OfB0TH2mLK4nlPjzO4T2s2m0VRdP2NBFGmJuZ0m73RHBMnGnt6gv7044fZbHY6na4PQxh7FovFbrerVIF2PGi12pgYP7tj9wkoCJOTk+UPQfkvsKurKy0tzeucefPmyT9MnTq1srJy3bp111xzTXJysnuAdXV1lZWV+fst8fHxX3/9dXJy76rACQkJXv9U90yFsccYMxgMob6KyHDO7Ly+RBUf7MEygiDo9frRCMLCBLFNYi02sTgtPlK6NkNCEASdTocgDCGVSqXRaAIPwkAE9HLm5+erVKpTp07JN48fP15SUjLA+fHx8VarlYgmTpx49uxZs9kcyAMZY0lJScl9gvvvBBhLp7oG2oApDOXF0ZEOHqcmpCBEoYCC0GAw3Hjjjb/61a+cTufevXs//vjj2267jYgqKyu/973vyeds2bKlvb2diL7++uvnnntuxYoVRDR16tSpU6c+/fTTnPO33nqrpaVl5cqVo/ZvAQgLNpGaLLzQEFF9hHHsm1aMlIEoFWiB//vf/76+vj4lJWXlypV//vOfCwoKiKirq2v79u3yCW+88caECRNiY2Ovv/76e++9995775WPr1u3btOmTQaD4dFHH3399de12oBXyweITJU9vNDA1BHVeJYbxw62ccymh+jEhjT8wdVNONQTBn0gEaWmplZUVKSkpCje29PTgz7C0DIajegjDMSmKumFk9J7lwd/vqzJZBqlPsIDbXzWv5y3lAgbL0bb6EBMJhP6CEPLbDaHpo/QZdC/QH8njMafLkB4quhS3u0onOXGMSKsuA1RCt9rAIKsonvwDZjCTZqWNCpCHyFEJwQhQJBVdPHSSKsIGVGOnqGPEKITghAgyCq6I2zuhCwvjrJQEUJUwvrXAMFkclKHjefFRV6ivLhYVRBRUz4AggVBCBBMp7t5cTwTIjBQJkZavyZAsKBpFCCYTkXactsAgCCEqHPPTvG8N53/b590smvwSbTNFvrRl2LgT366myZGYAchQDRDEELUOdXF75ssdNr5JR+Iszc5f3dYarX6PXlvK19fIQ3pySNuyChAlEMQQtSpMdHKQvaH+aqab6t/N0/1Sb305H6/Nd+Rdt5pJ6Mj0Cc/1cUnIwgBIgqCEKKLxKne3DuqU2B0cTb7yXTVwTa/baRHOjgR1ZoCXYnweCefnIQgBIgkCEKILk0WSo6lWLc3/owUdqjd75K7Rzp4ioZqTYE+uZpRGhaWB4goCEKILtVG7jVbLk1LcTGs2qgQhSKnii5+SY4QYEWIchAgEiEIIbrUmHi+z7TxGSl0qF0h6iq6eE4cm5QYaEV4opOfhyAEiDQIQhiH7P6HedaaKC/O++CMFHawTeHkox18ahLLi2MBVoQnulARAkQeBCGMN0YH5f7T0e1nnGeNief7rH82M4UpVoRHOmhaCg0hCDsxZBQg8iAIYbx5r1pqtVJlt3J0VRmpwGd34Rl+g5BPS2Z5cYE2jR7vpPOShna1ABByCEIYb946x7UqOu0nCGtNCitiT0pkNSZudnqf3BeEAVWEJie1Wb1H4gBA+EMQwrhidtL2Ouk/JgqV3conVBt5oU9FqBZoUiKTpwy62ESqNvLSRJamJYtIvjHp5UQnL02MyOW2AaIcghDGlfeqpQWZbG46U2wadUjUblPeh31mqnfr6LFOXpLA5BmHOfrBi0IMGQWIUAhCiDwnOvmT+5UHhr51lt9QJExMYIpNo3UmnqVjKqW08u0mPNrBpyb3nhpIN+GJLj4JI2UAIhCCECLPZ438yX3iiU7vqDM76aM6aVWhUBJPik2j1SbK92kXlc1IYV4LrckdhPLP+QF0E57ASBmAyIQghMhzspMXGJhvUbi5RpqXztK0lG9gLVZu9VlJu1Zp7oRsZgo73OGx0NqRdj4tuffngCpCLCsDEJkQhBB5TnTx/50rfFoveQ1veeMsv6lYICIVowIDO9vjXcNVK82dkKVpSadiNW4LrR3pIFdFmBvH6swDVYQip8oebMkLEJEQhBB5TnTS7DT20HTVUwf6i0Kzk7bWSqsKe9/SJQkKraOKs+ldZqTQwb5uwh4HtVr5hHhX0yjVGAe6pDPdPEfPtKoh/1sAIOQQhBBhrCI1WnhRPLt/ivBpvXS4L7q21Ehz01l6384PiuNlaoyU77O+msvMVHaovffnIx18SnL/XIhBpxIe7+ToIASIUAhCiDAnu3hJPFMxilPTw9NV/9NXFL55jt80of/9XBKvMINCccVtF/eBo0fa+fTk/jMHDcITXYTF1QAiFIIQIsxJtzEpP5gifNYgHW7nVpG21krXFroFYQKr9OkjrDEO3DTaP3D0aGf/3AkiStdRj4N8R9+4YKQMQORCEEKEOdFFk/saIePU9JMZql8ekLbUSLNSWYau/7SJPn2EZieZRUr1v2vuZLeF1o6082kp/cHGiHL0rM5/UYidCAEiF4IQIszJTo95698/T9jZKP36kHTjBI8384R4Vm3kTrcZFvIqowOElbzQ2tEOTkRHOvhUzz6/gWdQnMQGTAARC0EIEcZrzz+9mn4yXfV1C7+uyOPNrFFRho7VuNVwNaaBRsrIZqSwg+28xUoOiXI9G1EH6CZstJCaUapmqP8UAAgL6lBfAMAQcKJTPiuZff88ITGWsnTeJ09MoNPdNCG+9+bAHYQyecXRkgSPDkLZABUhVhkFiGioCCGS1Bh5UiyLj/E4qFPTdycpvJNLEjwGjiruROhFHjh6uJ1PT/ENQr8V4XEEIUAkQxBCJHEfKTMorxkUA8+ml81MYQfb+JEOPtUn2AauCNFBCBC5EIQQSbxGygxMbhp1UdyS10ualnRq2lrrMWRUlm/w6HF0hyAEiGgIQogkJ4YyOLM4gZ1xm0pYbaTC+AFO7zUzhVUbFfsIB2gaxb4TABEMQQiR5MRQKsLSBFbZ3b+hRCAVIRHNSGE5euY7BDRDS512svnMqTc5qd02eKMrAIQtBCFEkpND6SM0xFBcDDWaiYg67SQwSogZ7DFEs1LZjBSF4wKjbD2r99mD4kQnL0vsX5UUACIOghAiRreDuuwBVXUurqW3A5k7IbupWNh4ifK0IsXxMuggBIh0CEKIGPJImSFljmvgaLVp8LkTMhWjpFjluxS7CTF3AiDSIQghYgxppIzMtfR24BXhAPL0ShXhUFprASAMIQghYgxp7oTMNYMiwJEyA/OtCDnRN618ms8QUwCIIAhCiBjDqL2KE9gZuWk0gGVlBuXbR7ijgcfHEJpGASIaghAixolOPtTNb/sHywy4JW+AfCvCv5+QvjcZf0QAkQ1/wxAZRE5nenjpEIMwXUsipw5bQFtPDMqrImyz0ZYa6T9K8EcEENnwNwyR4Uw3z9YxrWrIDyyOZ6e7eZ2J5+pHWhFm6Vmbjdv79jhcVyFdUygkY/clgAiHIITIcKKLD29wZkkC29XEE2JJN+I9x1SMMnWsoW9O/QsnpXuUdr0AgMiCP2OIDCc6aXjz1icm0I6GoC2B5mod/aKJOyValIVhMgARD0EIkcF3P94ATUxg/26URj53QuYaL/P3E9I9k7GwGsB4gCCEyDCM2fSy4gTWbgvC3AmZXBF22endaumOUvz5AIwH+EuGyDCMuROyiQlERMFqGs3Vs1oTf+W0dHmukK4NylMCQIghCCECtNlI4pShG85jc/VMpw7C3AlZvoFqTfTCSekeTB8EGC/wxwwRYEjbEHoRGBXHs4IRz6aX5cWxT+qlbgddkoP+QYBxYsQjygFG3wi3Otq8XBW8wTLUbqOfzMAwGYDxA0EIEeBk15DXlHEXrHKQiLJ1LFlDq8vQlAIwfuDvGSJARReVJYT6IoiISC1Qw60xWcPqrQSA8IQghAhwamQVYXBphr7MGwCEMwQhhDuJ01kjL0kIlyAEgHEGQQjhrtrIUzUsDt3ZADA6EIQQ7iq6qSwx1BcBAOMXghDCXUUXL0W7KACMGgQhhLuK7jAaKQMA4w+CEMJdRRcvDY+5EwAwLiEIIdyd6iZUhAAwehCEENacEtUYeXE8ghAARguCEMLa2R6erWeYww4AowdBCGENcycAYLQhCCGsYe4EAIw2BCGENcydAIDRhiCEUdFgpqcPS3zEz4OKEABGG4IQRsU7VdIjX4n3fi6KIwvDim4qRR8hAIwmBCGMip2N/E8LVFU9/LYdokMa5pPYRKo388LgbasLAOALQQij4vMmfkUee2+52i7SdducFudwnqSyhxcaWAzepAAwmvAZA8F3roc7JF6SwGIFev1SVZqWXbnV2eMY8vNUdHHMnQCA0YYghODb2cQvyup9a6kYvbhYNSmRLd/iNA2xLqzopokYKQMAo2wIu53u2rVr3bp1giCsXr167ty5Xvc2Nja+9dZbBw4c0Ol011xzzbJly+Tj77777vHjx+WftVrtgw8+GJTrhnC2s5FflNUfYAKjv1yo+t5O8ebtzncuU6sD/vZV0cXLUxGEADC6Av1M2rt37xVXXDF9+vRJkyYtW7bs8OHDXic888wzu3fvvuCCC4qKir71rW+tXbtWPr5x48bdu3cH85Ih7O1s5IuzPAKMEf3lQpVGxVb/ewjDSE9h7gQAjL5AK8I//OEPP/zhD++//34iOnfu3DPPPPP3v//d/YRf/vKXgtAbq4Ig/OMf/7j77rvlm5dddtkPfvCD4F0zhLUWKzVb+LRk7wBTMdqwVLVsi/Pxb8QnZwe0eCjWVwOAMRBoRfjll18uXbpU/vniiy/etWuX9xMJ/U/V2tqalpbmuvnRRx/99Kc/Xbt2rcViGdHFQiTY2SgtzGSCUiGnU9O7l6tfP8OfPTr4jAqzk9ptPC8OFSEAjK5AK8LGxkZXtqWlpTU0NPg78+jRo88+++z27dvlm9OnT7darfHx8evWrXv66ae/+uorg8Gg+ECz2XznnXfGxsbKN2fNmvXwww+77rVYLCoV9iAIJbPZ7P51x59Pa1XzU8lstiveqyN6ewld9nFMmtq2Kn+gVtLDnawoTmW1mId5ueOR2WwmIsbw5SBkzGYz5zyQPwQYJWazWRTFwOMgNjZWrR4k6QINQo1GY7f3frTZbDadTqd42rlz56666qo//OEPc+bMkY/87Gc/k3946KGHZs2atX79+u9///v+Lve6665zxWRpaalWq3Xd63A43G/C2HM6nYG8BF+2Sn+cx7TaGH8nTNbSO8v4im3q8gxhkv91RGusvCyR40V3J4qiVqtFEIaQ/BIgCENIkiSNRhN4EAbyYgUahHl5eTU1NfPnzyeimpqa3Nxc33NqamouvfTSRx555Dvf+Y7Cb1Krp0+fXltb6/dS1Oprr702JSVF8V5BEPDmC61AXgKjg051iRdkqgc+cU4G/Wwm+/4u6dMVKn8f6hU90qSkgN7E0UN+CRCEIST0CfWFRK/ReAkCfa7rr79+w4YNRMQ5/+c//3n99dfLx99+++3W1lYiampquvzyy++99155QI2Mc97Z2Sn/3NDQ8Omnn55//vlBvHoIN18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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Test on random matrices\n",
    "proportion_list = zeros(100)\n",
    "step = .1\n",
    "\n",
    "for p = 1:100\n",
    "\n",
    "    # Generate new matrices\n",
    "    A = rand(5,5)\n",
    "    A_sym = (A+A')/2\n",
    "    A_skw = (A-A')\n",
    "\n",
    "    # Blend\n",
    "    X = skew_blend(A_sym, A_skw, p/100);\n",
    "    idx = asym_idx(X)\n",
    "    proportion_list[p] = idx\n",
    "end\n",
    "\n",
    "plot(proportion_list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create 10x10 matrix with Rank 4\n",
    "size = 10;\n",
    "tgt_rank = 4;\n",
    "\n",
    "A = randn(size, tgt_rank) * randn(tgt_rank, size);\n",
    "\n",
    "rank(A)\n",
    "\n",
    "# Create an mxn matrix with Rank r\n",
    "\n",
    "m = 10;\n",
    "n = 37;\n",
    "r = 4;\n",
    "\n",
    "B = randn(m, r) * randn(r, n);\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "false\n",
      "true\n",
      "true\n",
      "true\n",
      "true\n",
      "true\n",
      "true\n",
      "true\n",
      "true\n",
      "true\n",
      "true\n"
     ]
    }
   ],
   "source": [
    "# Test whether matrix rank is invariant to scalar multiplication\n",
    "\n",
    "for i = 0:.1:1\n",
    "    A = rand(10,10)\n",
    "    rank_a = rank(A)\n",
    "    rank_s = rank(i*A)\n",
    "    println(rank_a == rank_s)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10\n",
      "3\n",
      "10\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "Af = rand(10, 10)\n",
    "Ar = rand(10,3) * rand(3,10)\n",
    "l = 13\n",
    "\n",
    "println(rank(Af), '\\n', rank(Ar))\n",
    "\n",
    "println(rank(l*Af), '\\n', rank(l*Ar))"
   ]
  }
 ],
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