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  {
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   "metadata": {},
   "source": [
    "Chapter 04\n",
    "\n",
    "# diag() 函数\n",
    "Book_4《矩阵力量》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53c03812-0303-4691-9961-f87c9059705c",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "该代码首先定义了一个 $3 \\times 3$ 的矩阵 $A$：\n",
    "\n",
    "$$\n",
    "A = \\begin{bmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "然后，代码使用 `np.diag` 提取矩阵 $A$ 的对角元素，得到向量 $a = [1, 5, 9]$。接下来，通过向量 $a$ 构建了一个新的对角矩阵 $A_{\\text{diag}}$，其形式为：\n",
    "\n",
    "$$\n",
    "A_{\\text{diag}} = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 5 & 0 \\\\ 0 & 0 & 9 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "此过程展示了从一个矩阵中提取对角线元素并利用这些元素构造新的对角矩阵的操作。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ccb1e0a3-d0ed-4996-9880-b08ba4918993",
   "metadata": {},
   "source": [
    "## 导入所需库"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "e1227d8c-7f70-48b9-af13-a17776aa8152",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  # 导入NumPy库，用于数值计算"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49b8f8e3-3955-48f6-ab9c-de1db5afbe2a",
   "metadata": {},
   "source": [
    "## 定义矩阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c6358766-e0e7-480c-814e-c1ccfbae8e24",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 2, 3],\n",
       "        [4, 5, 6],\n",
       "        [7, 8, 9]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.matrix([[1, 2, 3],  # 定义矩阵A\n",
    "               [4, 5, 6],\n",
    "               [7, 8, 9]])\n",
    "A "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a682155-7c02-4961-acb9-ab7a3c1b17e8",
   "metadata": {},
   "source": [
    "## 提取对角元素"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "7b2ea9e5-48fa-4b8f-ad4c-6b040817a15d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 5, 9])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.diag(A)  # 提取矩阵A的对角元素\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4e2f66a-370e-410a-ab12-c1f071662dc9",
   "metadata": {},
   "source": [
    "## 构造对角矩阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "eea76f59-4071-47b4-b852-740f3df7045c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0],\n",
       "       [0, 5, 0],\n",
       "       [0, 0, 9]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_diag = np.diag(a)  # 使用对角元素a构造对角矩阵A_diag\n",
    "A_diag"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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