{
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   "source": [
    "Chapter 02\n",
    "\n",
    "# 逐项积\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": "ce4a4893-3f8c-402f-92c3-79d73b81d380",
   "metadata": {},
   "source": [
    "\n",
    "此代码定义了两个三维向量 $a$ 和 $b$，并计算了它们的逐元素积（即对应元素相乘）。代码中既使用了行向量形式也使用了列向量形式的 $a$ 和 $b$ 来演示逐元素乘法的计算。\n",
    "\n",
    "### 逐元素乘法公式\n",
    "对于两个向量 $a = \\begin{bmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{bmatrix}$ 和 $b = \\begin{bmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{bmatrix}$，逐元素积定义为：\n",
    "\n",
    "$$\n",
    "a \\odot b = \\begin{bmatrix} a_1 \\cdot b_1 \\\\ a_2 \\cdot b_2 \\\\ a_3 \\cdot b_3 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "代码中的计算展示了逐元素乘法的两种实现方式：使用 `np.multiply` 函数和直接使用 `*` 操作符。结果得到一个新向量，其中每个元素为对应元素相乘的值。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "370c070f-d704-40b0-90c9-ef2e2bee22ce",
   "metadata": {},
   "source": [
    "## 导入所需库"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b92befcd-0d09-4301-aecc-cb3172ce2cd9",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  # 导入NumPy库，用于数值计算"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25572211-3dd7-487d-ab16-d62682d66f56",
   "metadata": {},
   "source": [
    "## 定义两个行向量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "04b0208a-df81-48a4-8e85-f2ef347708c1",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([-2, 1, 1])  # 定义向量a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "557a8b24-f13d-4995-9bc2-c8bc3fa19389",
   "metadata": {},
   "outputs": [],
   "source": [
    "b = np.array([1, -2, -1])  # 定义向量b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a95a30f-b3fc-4e2f-b2e7-ccd49cc48ed1",
   "metadata": {},
   "source": [
    "## 计算行向量的逐元素积"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "29eca8ec-ed59-4e60-90bf-8405a4a084e3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2, -2, -1])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_times_b = np.multiply(a, b)  # 计算a和b的逐元素积\n",
    "a_times_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "91a581bf-ca56-44b2-8d21-8ae4a19ee309",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2, -2, -1])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_times_b_2 = a * b  # 使用*操作符计算a和b的逐元素积\n",
    "a_times_b_2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65964628-e9f9-4626-9a49-adc10d2f4ecf",
   "metadata": {},
   "source": [
    "## 定义两个列向量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "42eec7be-ccdc-4a9a-be68-0e074793c2a4",
   "metadata": {},
   "outputs": [],
   "source": [
    "a_col = np.array([[-2], [1], [1]])  # 定义列向量a_col"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b8757d8c-776a-400c-be13-0b7cac7ffdac",
   "metadata": {},
   "outputs": [],
   "source": [
    "b_col = np.array([[1], [-2], [-1]])  # 定义列向量b_col"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2269b17d-2fc2-47dc-b7eb-4f4ec992502f",
   "metadata": {},
   "source": [
    "## 计算列向量的逐元素积"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "c9f74e22-8432-4c40-bb9b-36f0a2d5e064",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2],\n",
       "       [-2],\n",
       "       [-1]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_times_b_col = np.multiply(a_col, b_col)  # 计算a_col和b_col的逐元素积\n",
    "a_times_b_col"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "7cd5c6c6-b2e3-4027-8eaa-7656594db66b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2],\n",
       "       [-2],\n",
       "       [-1]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_times_b_col_2 = a_col * b_col  # 使用*操作符计算a_col和b_col的逐元素积\n",
    "a_times_b_col_2"
   ]
  },
  {
   "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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