stuff

Author: alanedelman

.DS_Store

@@ -0,0 +1,107 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The jacobian of the symmetric matrix eigendecomposition is the vandermonde expression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 190,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "using LinearAlgebra, GenericLinearAlgebra, ForwardDiff"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$ dA = \\prod_{i <j} (\\lambda_i - \\lambda_j)(d\\Lambda)^\\wedge(Q'dQ)^\\wedge $$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 191,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n = 4\n",
+    "A = randn(n,n);\n",
+    "#A += A'  "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 192,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "upper = [i≤j for i=1:n, j=1:n][:]\n",
+    "Λ, Q =  eigen(Hermitian(A))\n",
+    "dΛ, dQ = (ForwardDiff.jacobian(t -> _eig(Hermitian(t)), A) for _eig=(eigvals, eigvecs))\n",
+    "dΛ, dQ = dΛ[:,upper], dQ[:,upper];"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 193,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "subupper = [i<j for i=1:n, j=1:n][:]\n",
+    "dQ = (kron(Matrix(I, size(A)),Q')*dQ)[subupper,:];  #compute Q'*dQ for every vec'd dQ"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 194,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(0.0007641193180814094, 0.0007641193180814138)"
+      ]
+     },
+     "execution_count": 194,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Experiment vs Theory\n",
+    "det([dΛ;dQ]),1/abs(prod([Λ[i]-Λ[j] for i=1:n, j=1:n if i<j]))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "@webio": {
+   "lastCommId": null,
+   "lastKernelId": null
+  },
+  "kernelspec": {
+   "display_name": "Julia 1.5.2",
+   "language": "julia",
+   "name": "julia-1.5"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.5.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}