Hw2 done

HW2.ipynb

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+    "### Ahmet Ercan Tekden\n",
+    "### 2012400159\n",
+    "\n",
+    "# Hw 2\n",
+    "\n",
+    "Vector and matrix p-norms are related by various inequalities, often involving the dimensions m or n. For each of the following, verify the inequality and give an example of a nonzero vector or matrix (for general m,n) for which equality is achieved. In this problem x is a m-vector and A is a m × n matrix."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "**(a)** $\\|x\\|_{\\infty} \\leq \\|x\\|_{2} $\n",
+    "\n",
+    "* (1) $\\|x\\|_{2} = \\sqrt{x_1^2+x_2^2+ ... +x_n^2} $\n",
+    "* (2) $ \\|x\\|_{\\infty} = max\\{|x_1|,|x_2|,..|x_n|\\} $\n",
+    "* (3) $ \\sqrt{x_1^2+x_2^2+ ... +x_n^2} \\ge |x_i|$ for all $ 1\\leq i \\leq n $\n",
+    "* (4) $ \\implies \\|x\\|_{\\infty} \\leq \\|x\\|_{2} $\n",
+    "\n",
+    "** ex: ** \n",
+    "\n",
+    "\\begin{equation}\n",
+    "    x =\n",
+    "    \\begin{bmatrix}\n",
+    "    1 \\\\\n",
+    "    2\n",
+    "    \\end{bmatrix}\n",
+    "\\end{equation}\n",
+    "\n",
+    "$ \\|x\\|_2 = \\sqrt{2^2+1^2} = 2.2360679775 $\n",
+    "\n",
+    "$ \\|x\\|_{\\infty} = max\\{2,1\\}= 2 $\n",
+    "\n",
+    "$ \\|x\\|_{\\infty} = 2 \\leq \\|x\\|_2  = 2.2360679775 $"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "**(b)** $\\|x\\|_{2} \\leq \\sqrt{m} \\|x\\|_{\\infty}$\n",
+    "\n",
+    "* (1) $ \\exists{j}$ where $|x_{j}|=max\\{|x_1|,|x_2|,..|x_n|\\} $\n",
+    "* (2) $ \\forall{x_i}$ we can say $x_i^2=x_j^2-b_i$ where $ 0 \\leq b_i \\leq x_j^2$ $\\implies \\|x\\|_{2}= \\sqrt{m*x_i^2 - \\sum{b_i}} \\leq \\sqrt{m}*x_i $ for any $x_i$\n",
+    "* (3) $ \\|x\\|_{2} \\leq \\sqrt{m}*x_i$ for any $x_i \\implies \\|x\\|_{2} \\leq \\sqrt{m} \\|x\\|_{\\infty}  $\n",
+    "\n",
+    "** ex: ** \n",
+    "using x in (a) ,\n",
+    "\n",
+    "$ \\|x\\|_2 = \\sqrt{2^2+1^2} = 2.2360679775 $\n",
+    "\n",
+    "$ \\sqrt{2} \\|x\\|_{\\infty} = 1.41421356237 * max\\{2,1\\}= 2.82842712 $\n",
+    "\n",
+    "$ \\|x\\|_2 = 2.2360679775 \\leq  \\sqrt{m} * \\|x\\|_{\\infty}  = 2.82842712 $"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "**(c)** $\\|A\\|_{\\infty} \\leq \\sqrt{n} \\|A\\|_{2}$\n",
+    "\n",
+    "If we multiply A with n-vector y we get Ay which is a m-vector. From here we can get following 2 equation.\n",
+    "\n",
+    "* (1) $\\|Ay\\|_{\\infty} \\leq \\|Ay\\|_2 $ using (a)\n",
+    "* (2) $\\|y\\|_2 \\leq \\sqrt{n} \\|y\\|_{\\infty} $ using (b)\n",
+    "\n",
+    "If we multiply equations we get \n",
+    "\n",
+    "* (3) $ \\|Ay\\|_{\\infty} * \\|y\\|_2 \\leq \\|Ay\\|_2 * \\sqrt{n} \\|y\\|_{\\infty} $\n",
+    "\n",
+    "if we divide both sides with $ \\|y\\|_2 * \\|y\\|_{\\infty} $ we get following equation\n",
+    "\n",
+    "* (4) $ \\|Ay\\|_{\\infty} \\mathbin{/} \\|y\\|_{\\infty} \\leq \\sqrt{n} * \\|Ay\\|_2 \\mathbin{/} \\|y\\|_2 $\n",
+    "\n",
+    "since $ \\|A\\|_2= \\|Ay\\|_2 \\mathbin{/} \\|y\\|_2 $ and $ \\|A\\|_{\\infty}= \\|Ay\\|_{\\infty} \\mathbin{/} \\|y\\|_{\\infty}  $\n",
+    "\n",
+    "we can see that (4) becomes\n",
+    "\n",
+    "* (5) $\\|A\\|_{\\infty} \\leq \\sqrt{n} \\|A\\|_{2}$\n",
+    "\n",
+    "** ex ** \n",
+    "\n",
+    "\\begin{equation}\n",
+    "    A =\n",
+    "    \\begin{bmatrix}\n",
+    "    1 & 3 & 4\\\\\n",
+    "    2 & 1 & 1\n",
+    "    \\end{bmatrix}\n",
+    "\\end{equation}\n",
+    "\n",
+    "\n",
+    "$ \\|A\\|_2=  5.4271 $\n",
+    "\n",
+    "$ \\|A\\|_{\\infty}= 8.0000  $\n",
+    "\n",
+    "$ \\|A\\|_{\\infty} = 8.0000 \\leq \\sqrt{n} \\|A\\|_{2} = 1.73205080757 * 5.4271 = 9.40001293776 $\n",
+    "\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "**(d)** $\\|A\\|_{2} \\leq \\sqrt{m}\\|A\\|_{\\infty}$\n",
+    "\n",
+    " just like (c) buth we use following equations instead\n",
+    " \n",
+    "* (1) $\\|y\\|_{\\infty} \\leq \\|y\\|_2 $ using (a)\n",
+    " \n",
+    "* (2) $\\|Ay\\|_2 \\leq \\sqrt{m} \\|Ay\\|_{\\infty} $ using (b) (Ay is m-vector unlike y which is n-vector)\n",
+    " \n",
+    "If we multiply equations we get \n",
+    "\n",
+    "* (3) $ \\|Ay\\|_2 * \\|y\\|_{\\infty} \\leq  \\sqrt{m} * \\|Ay\\|_{\\infty}  \\|y\\|_2 $\n",
+    "\n",
+    "if we divide both sides with $ \\|y\\|_2 * \\|y\\|_{\\infty} $ we get following equation\n",
+    "\n",
+    "* (4) $ \\|Ay\\|_2 \\mathbin{/} \\|y\\|_2 \\leq \\sqrt{m} * \\|Ay\\|_{\\infty} \\mathbin{/} \\|y\\|_{\\infty} $\n",
+    "\n",
+    "since $ \\|A\\|_2= \\|Ay\\|_2 \\mathbin{/} \\|y\\|_2 $ and $ \\|A\\|_{\\infty}= \\|Ay\\|_{\\infty} \\mathbin{/} \\|y\\|_{\\infty}  $\n",
+    "\n",
+    "we can see that (4) becomes\n",
+    "\n",
+    "* (5) $\\|A\\|_{2} \\leq \\sqrt{m}\\|A\\|_{\\infty}$\n",
+    "\n",
+    "** ex ** \n",
+    "\n",
+    "using same A as (c)\n",
+    "\n",
+    "\\begin{equation}\n",
+    "    A =\n",
+    "    \\begin{bmatrix}\n",
+    "    1 & 3 & 4\\\\\n",
+    "    2 & 1 & 1\n",
+    "    \\end{bmatrix}\n",
+    "\\end{equation}\n",
+    "\n",
+    "\n",
+    "$ \\|A\\|_2=  5.4271 $\n",
+    "\n",
+    "$ \\|A\\|_{\\infty}= 8.0000  $\n",
+    "\n",
+    "$ \\|A\\|_{2} =  5.4271  \\leq \\sqrt{m}\\|A\\|_{\\infty}  =  1.73205080757 * 8.0000 =  13.8564064606 $\n",
+    "\n",
+    "\n",
+    "\n"
+   ]
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