{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a88398a0-03e1-458f-ba48-518fb151b1b3",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "deac13e6-b3dd-4c8d-9c15-504052e60c6d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       " 0.693988  0.0373642  0.306196\n",
       " 0.475249  0.0523086  0.525352\n",
       " 0.275428  0.0212671  0.431897"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(3,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a7cfa41c-b406-475b-8a8e-bc56cab68cfa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}\n",
       "U factor:\n",
       "3×3 Matrix{Float64}:\n",
       " -0.649854   0.735055   0.193349\n",
       " -0.624454  -0.371326  -0.687149\n",
       " -0.433297  -0.567284   0.700316\n",
       "singular values:\n",
       "3-element Vector{Float64}:\n",
       " 1.1253009779161582\n",
       " 0.27878010490969024\n",
       " 0.013851237498536341\n",
       "Vt factor:\n",
       "3×3 Matrix{Float64}:\n",
       " -0.770554   -0.0587937  -0.634658\n",
       "  0.636347   -0.0144316  -0.771268\n",
       "  0.0361865  -0.998166    0.0485336"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, S, V = svd(A) #singular decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c6fdc076-bfc8-46e9-97c2-5427f4c49395",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 BitMatrix:\n",
       " 1  0  0\n",
       " 0  1  0\n",
       " 0  0  1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "3×3 BitMatrix:\n",
       " 1  0  0\n",
       " 0  1  0\n",
       " 0  0  1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "3×3 BitMatrix:\n",
       " 0  0  0\n",
       " 0  0  0\n",
       " 0  0  0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(U'*U .> 0.00001)\n",
    "display(V'*V .> 0.00001)\n",
    "display(A - (U * Diagonal(S) * V') .> 0.00001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "37b7a415-d7ea-4d00-b704-7d79e562d899",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " 0.019506810588402184\n",
       " 0.2433655332706062\n",
       " 0.9153216003851385"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " 1.1253009779161582\n",
       " 0.27878010490969024\n",
       " 0.013851237498536341"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(eigvals(A))\n",
    "display(svd(A).S)\n",
    "# singular values are more spread out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "5a4893c6-074c-4479-ac8a-9ad8db295d0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((3, 3), (3,), (5, 3))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "((5, 3), (3,), (3, 3))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# svd exists also for rectangular matrices\n",
    "U, S, V = svd(rand(3, 5))\n",
    "display((size(U), size(S), size(V)))\n",
    "U, S, V = svd(rand(5, 3))\n",
    "display((size(U), size(S), size(V)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "380565b8-f15d-4f3f-9a0f-85fc82809bdd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " 2.302775637731995\n",
       " 1.3027756377319948\n",
       " 0.0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "svdvals([2 0 1;0 1 1; 0 0 0; 0 0 0]) # rank is 2 -> so 2 non zero values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "deb05175-6a64-4c40-ab5c-e614ef85797f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}\n",
       "values:\n",
       "3-element Vector{Float64}:\n",
       " 4.440892098500626e-15\n",
       " 1.697224362268007\n",
       " 5.302775637731994\n",
       "vectors:\n",
       "3×3 Matrix{Float64}:\n",
       " -0.333333   0.444872  -0.831251\n",
       " -0.666667  -0.734656  -0.125841\n",
       "  0.666667  -0.51222   -0.541467"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       " 4.0  0.0  2.0\n",
       " 0.0  1.0  1.0\n",
       " 2.0  1.0  2.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [2 0 1;0 1 1; 0 0 0];\n",
    "display(eigen(A' * A))\n",
    "U, S, V = svd(A);\n",
    "display(V * Diagonal(S)' * Diagonal(S) * V')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "60516aff-16aa-41e6-a56d-6cad04547e55",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       " 1.0  -1.0  0.0\n",
       " 0.0   1.0  0.0\n",
       " 0.0   0.0  0.5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       "  1.0          -1.0  0.0\n",
       " -1.11022e-16   1.0  0.0\n",
       "  0.0           0.0  0.5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [1 1 0; 0 1 0; 0 0 2];\n",
    "display(inv(A))\n",
    "F = svd(A);\n",
    "display(F.V * Diagonal(F.S .^ -1) * F.U') # we can get the inverse from the svd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "cd9e606f-957d-4803-9ab8-f7b23c455e13",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2.0, false)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\"frobenius norm: 2.6457513110645907\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display((opnorm(A), opnorm(A) == norm(A))) # use opnorm for the matrix norm\n",
    "display(\"frobenius norm: $(norm(A))\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "9c469ac2-4d2e-4eba-9c6e-c63ec1fc06ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Matrix{Float64}:\n",
       " 1.27357  1.80721\n",
       " 2.87898  4.08529"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Eckhart-Young theorem\n",
    "# the min X with respect to opnorm(A-X) with rank less than or equal to k is:\n",
    "A = [1 2;3 4];\n",
    "F = svd(A);\n",
    "X = F.U[:,1] .* F.S[1, 1] * F.V[:, 1]';\n",
    "display(X)\n",
    "display(rank(X))"
   ]
  }
 ],
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