update 9.24

notes/2.ipynb

@@ -0,0 +1,137 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "7fc797ce",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "plt.rcParams[\"font.family\"] = \"D2Coding\""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a869c82a",
+   "metadata": {},
+   "source": [
+    "# Linear Equation\n",
+    "\n",
+    "$$A\\cdot x = b$$\n",
+    "\n",
+    "* $A$: [m, n]\n",
+    "* $b$: [m, 1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3bed3d44",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a5239ed9",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8bd218a8",
+   "metadata": {},
+   "source": [
+    "# SVD(Singular Value Decomposition)\n",
+    "\n",
+    "**why?**\n",
+    "\n",
+    "**what?**\n",
+    "\n",
+    "$$A = UWV^T$$\n",
+    "\n",
+    "* $U$: [m, n]\n",
+    "* $W$: [n, n] diagonal with positive or zero\n",
+    "* $V$: [n, n] orthogonal\n",
+    "\n",
+    "\n",
+    "**properties**\n",
+    "\n",
+    "* orthonormality\n",
+    "* uniqueness"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "563d7aec",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(4, 3)\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 1.,  0., -0.],\n",
+       "       [ 1.,  1., -1.],\n",
+       "       [ 0.,  1., -1.],\n",
+       "       [-1.,  1.,  1.]])"
+      ]
+     },
+     "execution_count": 26,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A = np.array([[1, 0, 0], [1, 1, -1], [0, 1, -1], [-1, 1, 1]])\n",
+    "print(A.shape)\n",
+    "U, S, Vh = np.linalg.svd(A, False)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fb89ad74",
+   "metadata": {},
+   "source": [
+    "## Homogeneous\n",
+    "\n",
+    "\n",
+    "## Nonhomogeneous\n",
+    "\n",
+    "\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "2025-02-Numerical (3.12.11)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.11"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}