Eigenvectors from eigenvalues (Julia demo)

Eigvec2fromeigval.ipynb

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    "# Eigenvectors from eigenvalues\n",
    "\n",
    "Short Julia notebook demonstrating the interesting property between eigenvectors and eigenvalues [announced](https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/) recently.\n",
    "\n",
    "![](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cv_%7Bi%2Cj%7D%7C%5E2+%5Cprod_%7Bk%3D1%3B+k+%5Cneq+i%7D%5En+%28%5Clambda_i%28A%29+-+%5Clambda_k%28A%29%29+%3D+%5Cprod_%7Bk%3D1%7D%5E%7Bn-1%7D+%28%5Clambda_i%28A%29+-+%5Clambda_k%28M_j%29%29&bg=ffffff&fg=000000&s=0)\n",
    "\n",
    "Inspired by a [Python notebook](https://github.com/leopd/geometric-intuition/blob/master/linear-algebra/eigenvectors%20from%20eigenvalues.ipynb), and by [Alan Edelman](https://twitter.com/AlanEdelman1/status/1195819794609659906)."
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   "source": [
    "using LinearAlgebra, RandomMatrices\n",
    "\n",
    "function eigvec2fromeigval(A)\n",
    "    n = size(A)[1]\n",
    "    λ = eigvals(A)\n",
    "    map((i,j) for j=1:n, i=1:n) do (i,j)\n",
    "        lhs = prod(λ[i] .- λ[1:n .!= i])\n",
    "        M = A[1:n .!= j, 1:n .!= j]\n",
    "        rhs = prod(λ[i] .- eigvals(M))\n",
    "        rhs / lhs\n",
    "    end\n",
    "end\n",
    "\n",
    "n = 5\n",
    "A = rand(GaussianHermite(2), n)\n",
    "v2 = abs2.(eigvecs(A))\n",
    "v2hat = eigvec2fromeigval(A)\n",
    "v2 ≈ v2hat"
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